BLOWUP PHENOMENA FOR THE VECTOR

NONLINEAR SCHRODINGER EQUATION

bY

James Coleman

-4 thesis submit ted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

@ Copyright by James Coleman 2001

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BLOPVUP PHENOMENA FOR THE VECTOR

NONLINEAR SCHRODNGER EQUATION

James Coleman

Doctor of Philosophy, 2001

Graduate Department of Mathematics

University of Toronto

We study various blomp phenornena associated with the vector nonlinear Schrodinger

(V'iLS) equation. This equation arises as a limiting case of the Zakharov system asso-

ciated with plasma physics. It is characterized by a positive parameter a which is related

to the mean thermal velocity of the electrons in the plasma. We are interested in studying

solutions tvhose HL-nom blows up in finite tirne. LVe show the existence of standing wave

solutions by solving a constrained minimization problem using the method of concentration

compactness. These standing waves are expressed in terms of ground state solutions of an

associated eUipcic boundary d u e problem. We numericaiiy construct ground States in both

two and three dimensions and analyze their structure. In the two dimensional case ive estab-

lish nurnerically that the limiting profile of bIonntp solutions of the W L S equation near the

blonnip point(s) is equd, up to rescaling, to the ground state. In the three dimensional case

we determine the blonrup rate as a h c t i o n of a. We deveIop a new dynamic mesh rehement

method to study time evohtion problems which b1ow up at more than one point, and apply

it to study soIutions of the W L S equation in which splitting of the profile occurs. Finaily,

Ne a p p l this new method ta study the time dispersion NLS equation, a perturbation of the

focusing XLS equation which arises in nonIineaz optics.

Acknowledgments

It is with enormous gratitude that 1 acknowledge the assistance of rny supervisor, Cather-

ine Sulem. who has been a constant source of encouragement. She generously donated many

hours of her time to provide me with guidance and support throughout the process of svriting

this thesis.

1 also rvould like to thank the rnembers of my committee, in particdar the extemal referee

Dr. RD. Russeii: as weII as Dr. James Coüiander and Dr. Robert Almgen, di of whom

provided vaiuable suggestions for improvements.

Additionaliyt 1 would like to thank the administrative and library sta$ of the department.

In particuiar, 1 arn gratefd to Ida Bulat for her warm and caring manner and constant

assistance in times of need.

1 also nrish to acknowledge the financial support which 1 have received £rom the Depart-

ment of Mathematics, the University of Toronto, and the Government of Canada-

Finaiiy, 1 wouici k e to thank my parents John and Barbara, my brother Gord, and

my sister Joyce, whose constant encouragement and mord support was so important to me

during the difficuit tirnes.

Contents

I . TWEORETICAL RESULTS

1 Introduction 1

1.1 Plasma dynamics and the Zakharov system . . . . . . . . . . . . . . . . . . . 1 1.2 The scalar nonlinear Schrodinger equation . . . . . . . . . . . . . . . . . . . 3 1.3 The vector nonlinear Schrodinger equation . . . . . . . . . . . . . . . . . . . 7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline 9

2 Ground states 12

. . . . . . . . . . . . . . . . . . . 2.1 Standing wave solutions and bound States 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Existence of ground states 13

2.3 A numericd method for eUiptic boundary-value problems . . . . . . . . . . . 20 39 2.4 Properties of the ground states (two dimensions) . . . . . . . . . . . . . . . . . .

2.5 Properties of the ground states (three dimensions) . . . . . . . . . . . . . . . 27

3 Asymptot ic structure of blowup solutions 3 1

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Esistence of b l o ~ u p soiutions 31 . . . . . . . . . . . . . . . . . . . . . . . 3.2 A blorcup resuit for the scalar case 32 . . . . . . . . . . . . . . . . . . . . . . 3.3 Variance identities for the vector case 39

II . MJMEFUCAL RESULTS FOR BLOWUP SOLUTIONS 4 Dynamic rescaling for single-peak solutions 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 SeIf-simiIar solutions 42 . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The method of dynamic rescaling 47

. . . . . . . . . . . . . . . . . . . . . 4.3 Single-point blowup in two dimensions 49 . . . . . . . . . . . . . . . . . . . . 4.4 Single-point blomp in three dimensions 52

5 A dynamic mesh refinement technique for singular solutions 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Multi-peak solutions 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Static mesh generation 64 3 Dynamic mesh rebement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Numerical results for dynamic mesh refinement 75

6.1 Single-point blowup for the NLS equation . . . . . . . . . . . . . . . . . . . 75 6.2 Two-point blowup for the W L S equation (a = 0.1) . . . . . . . . . . . . . . 78 6.3 Two-point bloivup for the W L S equation (a: = 0) . . . . . . . . . . . . . . . 81

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusions 86

7 The time dispersion NLS equation 90

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Physical derivat ion 90

7.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A Vector notation and inequalities 97

B Concentration compactness 99

C Coefficients for partial derivatives in curviiinear coordinates 106

References 108

Chapter 1

Introduction

1.1 Plasma dynamics and the Zakharov system

The vector noniinear Schrodinger equation, which we denote throughout this work as the

VYLS equation, can be viewed most naturdy as the subsonic limit of the Zakharov system.

In dimensionless variables, this system takes the form

where E : !El3 x R -+ IR3 is the envelope Eunction for the rapidly-oscillating component of

the electric field, n : R3 x B + W is the slowIy-mqing component of the particle density displacement. and a is a positive parameter.

These equations were h s t derived by Zakharov [84] as a mode1 of the large-space and

long-cime behavior of a Langmuir w e propagating through a charged plasma consisting

of two interpenetrsting gases, one composed of electrons and the other of protons. The

motion of these gases is assumed to be governed by the equations describing the behavior

of an inviscid fluid (namely the equation of continuity and Euler's equation). Since the

particles are charged they induce large-scale electric and magnetic fields Nithin the plasma,

which are described by Mauwell's equations. The complete systern is obtained by coupling

these two phenornena. An introduction to plasma dynamics and a heuristic derivation of the

Zakharov system cm be found in the reference by Dendy a more formai derivation of

these equations using multiple-scde anaiysis is given in Section 13.1 of [?II.

The dimensionless parameter (Y in (1.1) is given by

where c is the velocity of Light and v, is the mean electron thermal velocity. Since a contains

a factor of c' in the numerat or, it is quite large in typica1 situations. hdeed, its value ranges

fiom about 20 for labotatory plasmas to about 2 x IO' for interstellx gcts j7.11.

Mathematically, the Zakharov system hm associated with it several conserved quantities,

the most important of which are the mass or w e n u m b e r iV = I I E I I ~ ~ and the energy

( H d t o n i a n )

where Ii is the hydrociynarnic potential defined by &U = n + 1~1'. Using these conservation laws, one can derive an estimate [70] which Ieads to an wistence theorem which establishes

the existence of a local solution to (1.1) for sufllcientry smooth initial data, and global

existence if the initial conditions are also sufficiently small. This result was improved by

Added and Added [II in the special case a = 1 where the Linear operator appearing in the

first equation in (1.1) reduces to the LapIacian.

In some cases. one may wish to consider a simplification of (1.1) which corresponds

physically CO disregardhg the vectorial nature of the dectric fieId E and replacing it by a

scalar field .u. The parameter a then drops out and we obtain the scdar Zakharov system

iatu + AU = nu, &n - An = h]ul'.

One advantage of this formuiation is that it possesses physicdy mesiningful solutions which

are radiaily sjmmetric [151 and which consequently are easier to analyze rnathematically.

Other applications of the scaIâr system are given in [251. Note that, as with its vector

counterpart, the system (1.4) has consemed quantities such a s the rnass? energy, and linear

and m,&r momenta.

A n important property of the scdtlar systern is the presence of self-similar solutions. In

three dimensions such solutions can exist only asymptotically close to the biowup point when

the t e m An becornes negigible compared to the other terms in (1.4). In this iimiting case,

one can fomally construct solutions of the form [15, 29, 851

where U and !V are scalar functions on [O. a) satisfying

with appropriate boundary conditions. In two dimensions the term An can no longer be

negiected. and there exists an exact self-similar solution of the form

where now Cr and !V satisfy

1 AU - Cr - !VU = 0% 6' ($!v~ + 6qN, + 6iV) - AN = 4 (u') 4 = a, + -8,. (1.8)

rl

and p is a free positive parameter. Xo rigorous proof exists for the existence of non-trivial

solutions of (1.5-1.6), however for the two dimensional case (1.7-1.5) rigorous resdts have

been obtained by Glangetas and Uerle [29]. The equations for the profile have been studied

numericdIy in three dimensions by Zakharov and Shur [851 and in two dimensions by Bergé,

Dousseml Pelletier, and Pesme [5].

1.2 The scalar nonlinear Schrodinger equat ion

An important iimiting case of the scalar Zakharov system arises when one assumes that ntt

is negligible compared to An. Assuming that ,u and n decay at imînity, one may then take

n = so that (1.4) rediices sirnply to the nonlinenr Schredinger (NLS) equntion,

This equation also appears in many other physical contexts, such as the equation for the

envelope of a train of water waves in Auid mechauîcs [82]. As a sirnpIe example, we demon-

strate how it arises in a muitipIe scales analysis of the one-dimensional wave equation with

a noniïnearity of Klein-Gordon type,

if the nonlinear term is neglected, then (1.10) has traveling wave solutions of the form

.u(x, t ) = exp(i(kx - wt)) + c.c., where w and k are related by the dispersion relation

To study the effects of the nonlinear terrn. we let E be a small positive parameter and e-upand

the solution in terms of powers of E as

It is weU-known that if we attempt a regular perturbation expansion then the presence of

Iong time secular terms will Iead to breakdom. Hence' we use the method of multiple scales.

lntroducing the Iong distance and long time variables X = EX and Tl = d respectively, and

setting ,u = ,u(x: t . -Y, TI), we find to hs t order in e that

hom mhich n.e recover the same solution as in the Iinear case but now depending on the

variables X and Tl as nieIl,

u l ( x , t , :Yf Ti j = &(,Y, TL) exp ( i ( k x - ut)) T c.c., (1.14)

coupIed with the dispersion relation (1.1 1). At second order in E: ive find

The solvability condition for t his equation is

k where u, = ; is the group propagation speed. Equivalently. we can rewrite .iIi in terms of a

new function (; given by

Assuming (1.16) we t hen set .uz = 0.

To compute the third order term in the e-pansion (1.12) we introduce an additionai long

time variable T' = €9, and wite v = .u(x, t ! X, TL, q). The third order expansion of u then becomes

Substituthg this escression into (1.10) and equating coefkients of c3, we find

As the solvability condition for this equation we elirninate al1 of the h s t harmonic terms on

the right side by setting

which. up to a simple rescaling of the variables involved, is the cubic NLS equation in one

dimension. Assuming (1.20), u3 can now be evaluated e-qlicitly as

so that the fidi solution to (1.10) to order three in E is

cd - - $(E(x - u&)! É2t) exp (3 i (kx - wt)) + c.c.? 8

where p satisfies (1.20).

Turning now to the mathematical properties of the NLS equation: we first remark that

it is often written in the more general form

where the exTonent a satisfies

Xote that the upper limit disappears in the case d = 2. This restriction arises as a conse-

quence of the use of Sobolev embedding in the study of the Cauchy problem associated with

(1.23). If (1.24) holds then one can show that there exists a weak solution in HL(Rd) defined

on some maximal interva1 [O. t*) . If t* < m then we have [Iu(t)[lHi -+ ca as t -+ t*, we c d such solutions bIomp sohtions.

Equa~ion (1.23) aIso satisfies several important conservation laws. Specificaliy,. so long

as the sohtion is d e h e d in H1(Rd), the mass (or wavenumber)

N t ) = / lW12, the momentum,

and the energy (Hamiitonian) ,

are conserved. These conservation lam are related to a series of gauge transformations

under which (1.23) is invariant, which include spatial and temporal translations, conjugation

coupled nith tirne reversai, phase changes, spacetime dilations, and, in the case o d = 2,

pseud~conformai transformations.

-An important class of solutions to the NLS equation are standing wave solutions, which

are solutions of the Form ,u(x,t) = exp(it)R(lxl), where R satisfies the boundary Value

problem

Solutions to this equation are refened to as bound states. Among al1 bound states is one

in particular which minimizes a correspondhg action functionai, this is referred to as the

gound state, and plays a crucial role in the expression for the asymptotic form of blowup

solutions in the case a d = 2. This case is particuiarly important and is refened to as the

critical case. Similarly, the cases ad < 2 and a d > 2 are often described respectively as the subcriticd and supercritical cases.

A fundamental problem associated with (1.23) is to determine the conditions under which

the solution blows iip in H~(R"). The most important results in this area are that when

ad < 2 the solution always eicists globdy, while if ad = 3 the solution exists globaiiy if lluo[lL2 < IIRllLz? tvhere R is the ground state solution. The corresponding probIem of proving b lowp is more difficult. It is believed that whenever ad > 2 and H(uo) < O b lowp occurs, but this has been proven rigorously only in the cases where u o has finite mrimce

(l[xz~~(x) I l L 2 < MI) or where uo is radially qaimetric. in the case where blonrup is knom to occur, one c m characterize the nature of the solution

fairly explicitly, (see Section 4.1 for more details). Such solutions are characterized by a set

of points xi, . . . : x,, E Rd knom as biomp points, wïth the property that [u (x~ , t)l + as t - t*. For the criticai case, one can show that an amount of mass l [~ l l t~ equal to rhat of the gound state is concentrated near each point of blowup, and in fact the proNe of

the sohtion near a blowp point tends asymptoticdy to a rescaled version of the ground

state. Furthemore, the gowth rate of such solutions can be determined accurately, indeed

theoreticai and numerical analyses have concluded that

where t' is the bIosvup tirne. For the supercritical case. numericd simulations have s h o m

thût in the case of single-point b lowp the Mting profile near the blowup point Q is given

where Q : [O, CO) -. @ satisfies the boundary value problern

and a is a positive constant which appears to be independent of the initial condition u ~ . This

type of blowup solution is referred to as seIf-similar. For supercritical blowup solutions there

is no mass concentration phenornenon: indeed the amount of mass concentrated in each peak

decays to O near the blowup time.

1.3 The vect or nonlinear Schrodinger equation

In the previous section: we considered the subsonic limit of the scalar Zaliharov system and

obtained the scalar YLS equation. SimilarI~ one can take the subsonic limit of the vector

Zakharov system, which yieIds the vector nonlinear Schrodinger (WLS) equation

mhich will be the focus of most of the remainder of this work- Diagrammaticdy, we sum-

marize the logical relationship between the four systems considered so far in Figure 1.1.

1 VECTOR ZMHAROV SYSTEM 1 / '.

Subsonic Mt / \ Scalar limit / '.

VECTOR NLS EQUACON 1 1 SCALAR ZAKHAROV SYSTEM 1 ScaIar limit Subsonic limit

Fiove 1-1: ReIationship between the Zakharov system and the BIS equation.

7

As with the scalar NLS equation, we rviil ailow the nonlinear term of the VNL,S equation

to be siightly more general than a cubic nonlinearity. Additiondy? it is convenient to use

vector identities and make the substitution V x (V x E) = V(V - E) - AE. With these modifications, the VNLS equation takes the general form

where a > O and a satides (1.24). -4s with the scalar NLS equation, the most important case physicdly occurs when r = 1.

The Cauchy problern associated with the W L S equation can be handled using the same

techniques as in the scaiar case: in which one rewrites it in integal form and proves the

existence of Cived points by constructing a contraction mapping on a suitable Banach space.

This analysiç was carried out by Ginibre and Velo [26, 271 and later by Kato [.il! 421.

rUternative1y: one can prove existence by considering the VNLS equation as a h i t i n g case

of the Zakharov system and employing the estimates derived by Sulem and Sulem [70I. CVe

summarize the resdts as follows,

Theorem 1.1

For every Eo E H L (Rd), there exists rt weak solution E( t ) E H1(Rd) of (1.33) satiskng E(0) = Eo. This solution is defined on some maximal interval [O? t * ) where possibIy t* = co.

In adclition, the mass or wave number

the momentum

and the HamiItonian (energy)

are conserved in tirne-

The conservations lam @en above are reIated to various inVanance properties of the

VNLS equation- Specificalfyf if E is any solution of (1.331, then it is straightfomd to

ve* that so also are the functions obtained by spatial translation,

time translation,

Et(x, t) = E (x, t - to) , to E B,

conjugation coupled with time-reversal,

Et(x, t) = E (x, -t), (1.39)

phase changes,

E'(x, t) = exp(i0) E(x, t), û E B:

space-time dilation,

Et(x,t)=hLl"E(Xx,,\'t), X > O ,

and spatial rotations

Et(x. t) = 0 E (O-lx, t) . O E SO(R, d). (1.42)

These are the same invariances as in the scalar case except for the additional invariance

(1.42) due to the rotation goiip SO(R, (1). In the critical case ad = 2 we also have the

pseud~-conformai invariance

i~ lx l? Et(x, t ) = (rl+~t)'"!'exp ( 4(A + Bt) ) E ( x C i D t ) . A + & ' A+Bt . ID-BC=L(l-13)

discovered independently by Ginibre and Velo [28] and LVeinstein [78].

In the two-dimensional case, there is an additional duality between solutions correspond-

hg to reciprocd values of a. If E = (EL(xL, x?, t), E2(xLr x2, t)) is a solution to (1.33)

corresponding to some a? then by direct substitution, it is easy to verify that the Function

is a solution corresponding to alL.

In this work we adi be interested mainly in bIomp solution of the VNLS equation, that is

solutions for which the maximai existence time t* defhed in Theorem 1.1 is finite- For such

solutions nie have [[E(t) I I H L + oc as t -t t*. Vie are interested in particular in dete-

under what conditions blomp occurs, and in the case that it does occur. what the asymp

totic form of the solution is. We analyze this problem using both analytical and numerical

techniques.

We begin in Chapter 2 by studying the gound states (bound state solutions of minimal

action) associated with the WLS equation, which arise during the analysis of standing wave

solutions. We show the existence of such gound states by solving a variational problem.

In the course of the proof, we encounter the Fundamentai problem which distinguishes the

vector case from its scalar counterpart, namely the lack of radiai symmetry of the minimiz-

h g sequence. Because of this, the conventional techniques used to prove the existence of

minirnizers are inapplicable. Instead, we use the concentration compactness method fbst de-

veloped by Lions. LVe numericaiiy construct ground states in both two and three dimensions

for a range of values of o and describe their properties qualitatively and quantitatively.

In Chapter 3 we turn to the problem of existence of blowup solutions. The conventional

approach to the problem of proving blomp is to anaiyze the time evolution of the variance.

We formdate a general variance identity for the scaiar NLS equation which is mlid for

arbitrary weight functions and use it to present a simplified version of a blowup result

recently derived by Yaiva. We also extend this result to the supercritical case. Next, we

extend this variance identity to the vector case, which yields a blowup result in the speciai

case a = 1. Vie describe the dficulties encountered in attempting to generalize this result

to arbitrary values of a.

To study blowup solutions numericaiiy we turn in Chapter 4 to the method of dynamic

rescaling. After revieiving the theory involved, we perform a series of simulations in both

two and three dimensions with various values of a. In two dimensions, we verfi that the

asymptotic profile of the solution near the point of bloivup resembles the gound state R up to rescaiing, and we anai-yze the blomp rate of the solution. We End numericar evidence

that a log -log type blomp rate holds for aii values of a , but with a coefficient which varies

as a function of a and is equal to sr for a = 1, in accordance with the behavior in the scalar

case. In three dimensions* we anaiyze the behavior of the blowup rate a = -LL,, (where

L-i = 11E1IL..) and find that it is independent of the initiai conditions and hence depends

o d y on a. LVe determine the nature of this dependence numericaiiy.

In the course of these sirnuIations we encounter the phenomenon of splitting, in which a

solution with an initidy single-peaked profiie divides into two separate peaks as its amplitude

increases. The method of dynamic rescaling in its present form cannot be used directly

to study multi-peak solutions. To overcome this difEcdty, we derive in Chapter 5 a new

method for modeiing t his type of blomp. This method involves constnicting a curvilinear

mesh which adapts dynmically to the Eunction as it evolves by concentrating mesh points

near regions of high amplitude. The equations governing the time adaption of the mesh are

similu to those associated with the well-knowa Winslow method.

In Chapter 6 we use this new method to mode1 solutions of the WLS equation in which

splitting occurs. For smaU values of oz we observe that bIomp occurs at two points, and that

the solution riex these points is equal to the ground state up to rescaling. ive also consider

the k t k g case a = O in which blowup appears to take a different form. In this case the

gradient norm I[VE1IL2 blows up, but the divergence norm [IV . EllL.r remains bounded, while the amplitude increases very slowly and possibly saturates.

Finally, we employ our method in Chapter 7 to study a problem of nodinear optics as

sociated with dispersion of an dtra-short puise passing through a nonlinear medium. The

problem can be formulated mathematicaily as a non-eiiiptic NLS equation in three dimen-

sions. Previous analyses have suggested that the normal time dispersion term appearing in

this equation c m cause multi-splitting and a saturation of the amplitude. We ver@ that

this behavior occurs and construct profiles of the solution.

In Appendiv A we summarize basic notation and inequdities, in Appendk B -ive review

the method of concentration compactness as it is appbed in this paper, while in Appendix

C ive tabdate some coefficients used in our numericd simulations.

Chapter 2

Ground states

2.1 Standing wave solutions and bound states

The simplest Family of non-trivial solutions of the wLS equation are tirne-periodic solutions

of the form

E(x, t) = exp(iwt) R(x), (2-1)

where r! > O and R is real-dued function on Rd. By rescaling E, x, and t nre may as well talie the Erequency w to be unity; this assumption will be made hom now on. Sotutions

of this form are ccmmonly referred to as standing waves. Substitution of (2.1) into (1.33)

implies that R must solve

tvhere we assume R E H ' ( R ~ ) satisfies this equation in the weak sense. We call solutions to

(2.3) bound states. Note that there is actuaily a family of such equations dependhg on the

parameten o, d. and a. As usual? ive a s m e o E (O, A). A s d a r situation is encountered when studying the scalar NLS equation, nrhere the

bound state equation corresponding to (2.2) is

Solutions to (2.3) may be characterized as criticai points of the action functiond

defined on HI(R~). The analysis of the ground state problem for the scalar case has been

carried out by various authors, we summarize the most important results in the following

theorem.

Theorem 2.1

Suppose d > 2 md o E (0, A). Then (2.3) has an Wty of sphericdy symmetric solutions in C'(Rd) ivhich decay exponentially a t infini& which we deno te by R,, for n = 0,1,2! . . .. Each h c t i o n R, has e'ractly n zeros as a function of r = 1x1. Moreover, the function Ro

minimizes the action S arnong ail solutions of (2.3) and is the d q u e positive sohtion to

(2.3) up to translation. FinaUy, we have S(R,) - cx as n + m. The function Ro (the minimum-action bound state), is referred to as a gound state

for (2.3) and is usually denoted simply as R. The existence of ground states for equations

of the type (2.3) was s h o ~ n by Strauss [691, later Berestycki and Lions [?, 81 proved that

there are idinitely many bound states. Both of these results were obtained by considering a

constrained minimization problem. The existence of ground states can be s h o m by solving

an unconstrained minimization problem as is done by Weinstein [77]. Nternatively, one

c m use the approach of Jones, Kipper, and Plakties [4O] and of Grillakis [34] who develop

methods for proving the existence of radial solutions with a prescribed number of nodes.

To prove uniqiieness of R, one uses the fact that the ground state is positive and radially

symmetric with respect to some point in IRd to reduce the problem to analyzing an ODE.

This was resolved by Coffman [21] and McLeod and Serrin [58] under certain conditions on

O and d, md finally soIved in the general case by Kwong [441.

2.2 Existence of ground states

The extension of these results to the vector case is not straightforward. The f i c u i t y essen-

tially arises fiom the fact that? in contrast to the scalar case, we can longer assume that min-

imizers of variationai problems associated with (2.2) are radialiy symmetric. Consequently,

we are unable to exploit the compactness of the embedding of H&(Ed) in L2=+' (Rd),

the usuai method of proof breaks d o m .

These difficuities c m be overcome through the concentration-compactness approach de-

veloped by Lions [52, 531. -& with the classicai approach, concentration-compactness is

formulated in terms of a minimization problem for some appropriate functional. The basic

functionals which ive consider are the mass, kinetic enerm! and potential energy, defined by

respectively In terms of these quantities the total energy (Hamiltonian) and action h c -

tionals are given by

Formally, solutions of (2.2) correspond to critical points of the action S, but since this

functional is unbounded from below it is more usefui to consider some type of constrained

minimization problem. For example, one can characterize ground states a s minimizers of the

energy for fived mass,

as in done in the scaiar case when studying the dynamical stability of groiuid states [l?, 791.

Xote hoivever that this rninimization problem is weli-posed oniy if ad 5 2.

In analyzing the bound state problem associated with (2.2), Colin and Weinstein [22]

considered two other variationai formulations, namely the doubly-constrained minimization

problem

I ( X . p) = inf (-V(E)) , iV(E)=X, T(E)=p

and the unconstrained minimization probIem

1 = inf J (E) , E#O

both of which are vaüd for O < O. It foilom immediately from (2.5) and (2.9) that the quantities IV, T, V, and J scale as

respectiveIy. In par t icdu. J is invariant under this mapping, so that if E minimizes J then

so does Ex,, for any X and p. Hence, by replacing E with Ex,, for appropriately chosen

X and p we rnay as weil assume that N(E) = T(E) = 1. It then foilows £rom (2.9) that

V(E) = I-', so that (2.10) simplifies to

a further rescaling of the form (2.11) can be used to normalize the constants in hont of the

three terms on the Ieft side of (2.12) to be uni& and we then obtain a minirnizer of (2.9)

that solves (2.3).

The scalar version of (2.9) is

and in fact. the esistence of ground state solutions for (2.3) can be proven ciirectly by rnini-

rnizing (2.14) as is done by Weinstein [TI. The vector case is somewhat more complicated,

and in [22] the authors establish the existence of minimizers for (2.9) by solving the equiv-

dent probIern (2.5), although the details of the proof are omitted. They also prove several

other resdts concerning bound states.

In this chapter. we estabiish the existence of bound states as minimizers of the constrained

variational problem

I (X) = id (T(E) +!V(E)) = inf V(E)=A V(E)=X

(/ ( Q V E ~ ~ + (L - ujlV E[' + EI') Before stating the main resdt! ive begin by presenting several lemmas. The first one

charac terizes the minimum value attained in (2.15) in terms of the value of the constraint X.

Lemma 2.2

For the minimization problem defined by (2.15) we have I(A) > 0, and moreover I(X) = h = k ( l ) -

Pro0 f

Let X > O be arbitrary Then, if V(E) = A, we have by the Gagliardo-Wienberg inequality (A.9) that

2af 2 2*(2-&qE)ad/2 = I IEI~~=+~ r CIIEII~.,

where C is independent of E. It foilows that

[ ( A ) = inf (T(E) + !V(E)) 3 inf V(E)=X EFO

which proves the first result. Yex*, we note that

and consequently

which proves the second resuit. I

Corollary 2.3

For X > O and I (A ) defineci in (2.15), we have

Pro01

Since ~7 > O we have that I ( X ) is concave d o m as a function of X . Hence, for any X > O and p E (Of A) we have

Adding these inequalities gives the required result. I

The nekt lemma is simply the vector analogue of the classical Pohozaev equaiîties [65].

Lemma 2.4

I f E E H L (IRd) is a solution of (2.2) , then we have

Proo f

kIdtiplying (2.2) by Ë and integrating over Eld, we obtain

Similarly, multiplying (2.2) by (x v)Ë and integrating over Rd, we obtain

Combining the above equations and using (2.5) gives the desired resdt. I

Lemma 2.5

There exkits some X > O such that any minimizer of (2.15) is a gound state soiution of (2.2).

Prao f

Let E be a minimizer corresponding to some X > O. By the theory of Lagrange multipliers, there e'cists some c E R such that

in the weak sense in lYL (IRd). Muit iplying (2.16) by Ë and integrat ing over !Rd gives T(E) + N(E) = cV(E), and so

where we have used Lemma 2.2. Hence, any minirnizer of (2.15) correspooding to X = 1(1)*

is also a solution of (2.2) . Now let Et be any solution of (2.2) and let At = V ( E t ) . Rom

Lernma. '2.4 we have T(E) + N(E) = V(E) and T(Et) + !V(Et) = V(E1), and so 1

At V(Ef) T(E')+N(Et) I(At) - a i 1

-=-= > -= (;) ,\ V(E) T(E) + !V(E) - I ( A )

It follows that A' 2 A. Applying Lemma 2.4 again we have that S(E) = T(E) + N(E) - Iv(E) UYI = s V ( E ) . and similady S(Ef) = %V(E1). We conclude that S(Ef) > S(E), and so E minimizes the action S over al1 solutions of (2.2), as required. 1

We now turn to the main resuit of this section.

Theorem 2.6

For evey A > O there e.&s a minimizer of (2.15).

Pm01

The proof of the theorem is based on the method of concentration compactness due to

Lions [52, 531. In ilppendk B, we r e c d the main result of this paper in Lernma B.1, and

restate it in the form that is used in our conte* in Lemma B.2.

To show the existence of a minimizer for (2.15), fk X > O and let E, be any minimizing sequence. Then V(E,) = X and T(E,) + N(En) - I ( X ) . AppIying Lemma B.2, we see that there e,.cists a subsequence (aIso denoted by En) for which one of the following situations

occurs-

1. (Vanishing) For every R > O: we have

2. (Dichotomy) There exists p E (O, A) such that for every c > O there exist sequences EA and E', in HL(Bd) which for ail sdliciently Iarge n sati*,

3. (Compactness) There esists a sequence xn E Rd such that, for every e > O there exists R > O For which

The goal is to eliminate the possibility OF vanishing or dichotomy. It wiil then fol~ow that

concentration occurs. The possibility of vanishiig d be excluded through the FolIowing

lemma? which is similar to Lemma 1.1 in [53).

Lemma 2.7

Let p E (2, -&) , and let En be a sequence which js bounded in HL(Rd) . Suppose that

for some R > O. Then En - O in Lp(Rd) . Pro0 f

We use some of the ideas of the proof of Lemma 8.3.7 in [18]. Let ibf > O be such that llEnllH, 5 LW Eor aii n, and define a positive sequence E, by

Then by hypothesis 6, 4 O as n + m. Cover Rd by a sequence of cubes {Ci) such that

diam(Ci) = 2R and Ci n C, = 0 for i # j. Then for d i E N we have

since Ci is contained in some b d of radius R. Now, by Sobolev embedding we have

where C is independent of En and i. Surnming this inequaiity over i E N we get

and the result follows. I

Xow ive retuni to the proof of Theorem 2.6. First, suppose that vanishing occurs for

some siibseqitence En. Then. since En is a minimizing sequence we have in particular that

En is boiulded in H'(Rd). Hence, the conditions of Lemma 2-7 hold with p = 2 0 + 2, and ive conclude that En 0 in L2"+' (Rd), which contradicts the constraint V(E,) = X > 0.

Yext, suppose that dichotomy occurs. Fk E > O: and let ER and En be sequences satisfying (2.19) and (2.20). Since En is a minimizing sequence, ive have for sufficiently large

n that

T(E,) i- iV(E,) 5 I(X) + E. (2.22) From (2.19) we have V(EA) 2 p - E and V(E',) 2 (A - p ) - E. From Lemma 2.2 it foliows that [(A) is a continuous and increasing function of X , and combining these results implies

that there exists some C > O independent of E such that

Combining (2.20): (2,22), and (2.23) we get

Since E is a r b i t r q it follows that I(X) 1 I(p) + I(X - p) , which contradicts Corollary 2.3. Hence dichotomy does not occur.

We conclude that it is concentration that occurs. Thus, there evists a subsequence En

and a sequence x, E !Etd S I C ~ that (2.21) holds. Define a sequence É, by Ën(*) = En(.+xn)- Then, for eveq- E > O there exists R > O such that

Since Ën is bounded in H L ( g ) there exists some E so that fi, - E weakly in H ~ ( R ~ ) . UO, since B(0, R) is bounded, the injection HI (B(0 , R)) -+ LZuf2 (B(0, R)) is compact. Hence,

Ë. converges to E. strongly in L'~+' (B(0 . R)) for every R > 0, and it foiiows From (2.25) 2mf2 that V(E) = IlEl[ p.? = A- In addition, since T + N is weaidy lower serni-continuoust we

have

T(E) + N(E) 5 lirn (T(E,) + !V(E,)) = [ ( X ) . n-CE

(2.26)

Hence T(E) + N(E) = I(X), and so E is a rninirnizer for (2.15). 1 The problem of proving uniqueness of the vector ground state appears to be quite difficult.

The proof used in [44] for the scdar case depends cruciaiiy on the fact that R is radidy

symmetric. As we demonstrate in Sections 2.4 and 2.5 where we numericdy construct

ground States R for a range of values of a, the amplitude IR( is not radialIy symmetric

except when CY = 1. so that this method of proof cannot be casried over to the vector case.

Yote that when we speak of uniqueness for solutions to (2.2), we in fact mean uniqueness

up to the family of g a g e transformations under which this equation is invariant. Indeed, if

E is any solution of (2.2), then so are the functions given by

for B E R, O E SO(d, R), and x Rd. These transformations correspond respectively to

phase changes, rotatioris: and spatial translations, and are special cases of the more generd

family of invariances for the time-dependent WLS equation given in Section 1.3.

2.3 A numerical met hod for ellipt ic boundary-value problems

The bound state equation (2.3) is a non-linear, vector-valued elliptic boundasy d u e probIem.

For the corresponding scalar problem, one can restrict the search for solutions to sphericdly

symmetric functions, which reduces to solving the boundary value problem

This problem can be solved numericaily by a shooting method, as is done by Budd, Chen,

and Russell [13] and Budd [141 in a somewhat more general context.

For the vector case there has been comparatively little study. Since soltitions are no longer

radial one must work in the full space Rdl or at least in a sufEciently large d-dimensional

box The method which we describe in this section for numericdy solving (2.2) is based on

an iterative procedure which begins with an initial mess and converges rapidly to an exact

solut ion.

As previously mentioned, (2.3) is invariant under the family of gauge transformations

(2.27). The three degrees of Freedom associated with these invariances c m be eiimïnated if

we specify that solutions E to (2.2) m u t satisfy the normalization conditions

where X > O. The first equation specifies the choice of 9 and 0, while the second specifies the choice of x by requiring that the centre OF m a s of E be located at the origin. Note that

(2.79) implicitly assumes that bound states do not vanish a t their centre of mus. Al1 of the

bound states which ive construct satisfy this property.

Vie now describe the method in detail. FLxing a > 0, we want a solution E to the equation F(E) = O, where

which satisfies (2.29). We constmct this function through the method of quasi-linearization,

by d e h i n g a series of sipproximants Eo, EL, El:. . . which converge to a solution E, If En is one such approximant and w e set e, = E -En, then to leading order 6, satis6es Ln(%) =

F(E,), where

Ln(e) = uAe + (1 - a)V(V - e) - e + (2r7 + l ) l ~ $ ~ e . (2.31) This motivates the follotving iterative method. CVe begin by choosing some initial estimate

Eo (using a method which we describe Iater), and we then define a sequence of Functions

recursively by En = En-, ++-L for n 2 1, where e,+~ is determined Çom (2.31). tVe iterate

this process until for some n we have < E, where E is some fked tolerance. -4t this point En has essentiaiiy converged to a Eked point, which we take to be the desired solution

E.

The linearized differential equation Ln(+) = F(E,) is discretized using a tinite Werence

method. We work on the domain D = [-L, L ] ~ c Eld, where L is chosen to be sufEciently large, and use as approximate boundary conditions that E = O on bd(D). This boundary

condition is impIemented by choosing the initial estimate to satisfy Eo = O on bd(D), and

by solving the linearized equation with boundary condition e,, = O on bd(D) as weU. The

interval [O, LI is divided into n equal subdivisions. There are hence (2n - I ) ~ interior points

where we want to calculate the value of e,,. This @es a total of N = (d (2n - l) ld real

unknom which have to be calculated. The difFerential operators are appro-ximated by

a seven-point scheme in each direction (accurate to sixth order in h = L/n), and at the

b o u n d q are calcuiated by extrapoIating to fictitious points outside D where e,, is assumed

to vanish identicaily. The resulting linear system c m be expressed in the form Ax = b,

where x and b are vectors with LV elemezfts, and A is a sparse positive matriv with N2

elements. This system is solved by the conjugate-gradient method as described in [66].

We now consider the problem of choosing an initial estimate Eo. Suppose that we are

constructing a ground state R, namely a bound state of minimum action. For a = 1, the

ground state which satisfies (2.29) is given by R = (R, O, - - - ,O), where R is the ground state for the scalar NLS equation which we compute by a shooting method. To find Eo for

a # 1. we make the assumption thstt R varies continuously as a function of a. This means that the ground state R corresponding to a = I should be a good approximation for ground

states R corresponding to values of a sufficiently close to one. Hence, for such d u e s we

cake Eo = (R: 0:. . . : 0). LVe found e'rperimentalIy that this approximation is mlid? indeed for al1 0.5 5 u 5 2.0 the iterative process converged to an exact numerical solution. By using a continuation process we can then extend our results to a large range of values of a.

2.4 Properties of the ground states (two dimensions)

In this section we study the properties of the numerically constructed ground states which

were obtained for the critical case d = 2 and o = 1. We used seven different values for a ,

ranging Erom 0.1 to 10.0. For ai i d u e s of ar we used n = 100, while for L we used d u e s

which varied between 5 and 15 depending on the value of a. The value of L was chosen as

foilows. In order to obtain maximum accuracy, L should be chosen so that enor induced

by restricting the domain to the set [-Lt LI' is as small as possible. This can be achieved by taking L to be sufEciently Iarge. On the other band, for fked n the mesh spacing h is

proportiona1 to L, so that the box size should not be made too large. To baiance these tnro

constraints, we took L to be srnallest possible d u e such that the amplitude of the solution

at grid points adjacent to the boundary was less than 1 0 ~ ~ .

The results are ,Pivert in two forrns. In Figure 2.1 we have tabdated the minimum and

maximum d u e s and the L"norm for both components Rt and Rz of R, while in Figures 22-27 we show surface plots of these functions (except for the trivial case (Y = 1).

Figure 3.1: Properties of the numericdy computed ground state R for (u, dd) = Cl,?).

For a = 1, ive obtained a maximum amplitude of 2.306, which is in agreement to three

decimal places with the k n o m d u e computed in the scalm case [13]. In "ew of the

discussion given above on the method for choosing the side Iength L, it is reasonable to

estimate the mmimum error for these computations to be on the order of IO-^. For ix < 1: ive observe that the contours of Rt are roughly eiiiptical, with the major axis

of the ellipse aligned along the xL-zxxis and the minor ~ x i s dong the x p - a - . -4s a decreases

t o w d s zero, these contours become more elongated, and eventuaily assume a dipolelike

structure. For dl values of a sufficiently close t u one, Rl is strictly positive, however as CY

decrenses it eventually develops a pair of minima at the points (0, *c ) (where c > O depends on a): at which it assumes a negative value. Rom Figure 2.1 the threshold vaiue of CY for

the onset of this behwior appears to be near 0.2. For R2 we observe local maxima in the h s t and third quadrants, and local minima in the second and fourth quadrants having a

ma,.nitwie which increases as a moves away from 1. Findy, as û tends to zero we End that

both Rr and R2 became more concentrated near the origin.

For a > 1 sirnilac phenornena were obsemed. In this case, the contours of RL are again eUipticd3 but with the major axis aiigned along the x2-axis and the minor axis dong the

q-a&. Again, as a becomes very large the contours assume a dipule structure, and RI

develops two global minima on the xi-a.es near a = 5.0. For the component Ra we observe

local minima in the fmt and third quadrants and local maxima in the second and fourth

quadrants, having a magnitude svhich increases as u becomes Iarge. Finaily, as cr tends to

infinity we 6nd that both RI and & becorne more dispersed away h m the origin. For all values of a: we observe the symmetry rdations

Figure 2.2: Surface plots of RI (Ieft) and R2 (right) for (a! d) = (1,2) and a = 0.1.

Fi,gre 2.3: Surface plots of RL (Ieft) and R2 (right) for (u, d) = (1,2) and CI = 0.2.

Fiope 2.4: Surface plots of RI (lei31 and R2 (nght) for (a, d) = (112) and u = 0.5.

Fiorne 2.5: Surface plots of RI (left) and R2 (right) for (a, d) = (1,2) and a! = 2.0.

Figure 2.6: Siuface plots of Ri (lefi) and R2 (right) for (a, d) = (1,2) and ai = 5.0.

Figure 2.7: Surface plots of RI (Ieft) and R2 (right) for (CT, d) = (I,2) and a = 10.0.

for RI and R2 respectively. Consequently, the magnitude [RI is symmetric with respect to both the IL and x2-a~es, and in particular the centre of mass is situated a t the origin, as

expected. Finally, we note that there is a dichotomy between ground states correspond-

ing to reciprocal values of a. indeed, if R = (RI(x1, 1 2 ) , Rs(xl, ~ 2 ) ) is a ground state conesponding ta a particular value of ai, then the ground state for a-L is given by

This is of course to be expected in view of the invariance noted in (1.a).

FVhile the results presented above are interesting in their o m right, their primary signifi-

cance Lies in the relation with the structure of b lomp solutions. Indeed, ive wiiI demonstrate

nunierically in Chapters 4 and 6 that, in the case of two dimensions and cubic nonlinearity,

the profile of solutions near the blowup point(s) is equd to that of the ground state. This

extends the results observed for the scalar XLS equation ta the vector case.

2.5 Properties of the ground states (three dimensions)

In this section ive study the properties of the nrimerically constmcted ground states which

rvere obtained for the supercriticai case d = 3 and cr = 1. We used five difEerent values for

0, ranging €rom 0.2 to 0.0. For al1 values of a: we used n = 80? while for L we used values

which varied between 4 and S depending on the d u e of a.

FVe begin by giving some qualitative results. As expected from the normaiizat ion condi-

tion R(0, O! 0) = (A! O! O), the ground state R satisfies R(0x) = OR(x) for al1 rotations O

about the xl-axis. CVith respect to the plane xl = O we observe the symmetry relations

Rl(-x1,22:~3) =R1(~1,~2c:!,x3),

R2(-~1r~2ix3) = - R ~ ( X ~ ~ X ~ > Y X ~ ) T (2.35)

&(-XI, ~ 2 ~ x 3 ) = -&(EL, ~ 2 , ~ 3 ) ,

and, cornbiniag these two results, we summarize the symmetries which the component h c -

tions R I , R2, and RJ satisfy with respect to the variables XI, q, and x3 in Figure 2.8 below.

In particular. the amplitude h c t i o n IR[ is symmetnc with respect to aX three coordinate

planes and the centre of mass lies at the origin, as evpected on physicd grounds.

1 ;; 1 even 1 odd 1 even 1 even even odd

Fiogre 2.5: Symmetry propm-ties of the ground state R for (a, d) = (1,3).

In Figures 2.10-2.13 we show contour plots of the amplitude of R in both the longitudinal

direction (a cross-sectional view in the XI-x? plane, or equivalently in any plane containing

the xL-asis). and in the transverse direction (a cross-sectiond view in the Xa-x3 plane.) For

a < 1 the three-dimensional contours of IR1 resemble prolate eIiipsoids, nihile for a > 1 they resemble oblate ellipsoids. As a tends to O the proNe of the ground state becomes

increasingly elonpated in the xl-direction, whiIe as û: tends to infinity it becomes ffattened

and extends outward dong the x-ZQ plane,

Finaily, we tabulate in Fi,we 2.9 the minimum and ma-mum d u e s and L'-noms of

the thee cornponent functions. As expected. the components R2 and R3 vanish when cr = I

and increase in amplitude as a! tends to O or to inlinity. Notice as well that the syrnmetry

between ground states corresponding to reciprocal d u e s of a which we obsemed in the

two-dimensional case does not hold in three dimensions.

Figure 2.9: Properties of the numericdy computed gound state R for (a, d) = (1,3).

ac 0.2 0.5 1.0 2.0 5.0

RI min

-0.004 O O O

R2 m u 4.536 4.390 4.338 4.417

R3 L2-nom

1.568 2.922 4.345 5.738

-0.098 1 4.742

L'-nom 0.329 0.318 O

0-729

min -0.370 -0.212

O -0.275 -0.606

min -0.370 -0.212

O -0.270

6.997

max 0.370 0.212 O

0.375 -0.606

mau 0.370

0.606 1 2.005

P - n o m 0.329

0.212 O

0.275 0.606

0.318 O

0.739 2.005

-0.8 -0.6 -0.4 -0.2 O 0.2 0.4 0.6 0.8

(a) lonQtudinal view (rl-zl_ plane) (b) transverse view (52-x3 plane)

Figure 2.10: Contour plots of gound state amplitude IR[ for (a, d) = (1,3) and a = 0.2.

-1.5 II -1 4.5 O 0.5 1 1.5 -Is -1 -0.5 O 0.5 1 (a) longitudind view (ri-q piane) (b) transverse view (x2-x3 plane)

Figure 2.11: Contour plots of p u n d state ampIitude IR1 for (a, cl) = (1,3) and a = 0.5.

29

-zJ -1 O 1

(a) longitudinal view (xl-x? plane) (b) transverse view (--x3 plane)

Figure 2.12: Contour plots of gound state amplitude IR1 for (a, d) = (1,3) and a = 2.0.

-2 -1 O 1 2

(a) longitudinal view (xr-z2 plane)

-2 -1 O i 2

(b) transverse view (x2-z3 plane)

Figure 2.13: Contour plots of ground state amplitude IR[ for ( ~ , d ) = (1,3) and a = 5.0.

30

Chapter 3

Asymptotic structure of blowup solut ions

3.1 Existence of blowup solutions

The problem of proving the existence of bloivup solutions for the NLS and W S equations

goes back to the 1970's. For the scalar case, the h s t results were obtained by studying the

time evolution of the variance

Assuming finite variance, we obtain by differentiating this equation twice with respect to

time and using (1.23) the result [75]

which is often referred to as the variance identity or w i a l theorem. It foilows from (3.2)

that if ad 3 2 and H < O then blowup occurs in fibite time, in the sense that there exists some t* > O such that IIVu(t)lltz -+ cm as t + t*. If we assume super-criticality (ad > 2) then (3.2) can also be used to derive somewhat stronger results [451. Alternatively, in the

supercriticai case one can use the approach of GIassey [311 who considers the tirne derivative

of the tariance instead of the variance itself.

A major goal of recent research has been to extend these r e d t s to the case where the

variance is not necessarily finite. Various approaches have been taken in this direction. One

such r e d t , due to Ogawa and Tsutsumi [63], asserts that finite t h e blonnip always occurs

if uo is radidy qmmetric and the HamiItonian is negative. Their method is to use, instead

of lx]', a weight tünction which is bounded at ïnfinity. A crucial eIement of this proof is

the classical lemma of Strauss [69] which is used to control the behavior of u at idhity. A

similar arement vas used by Uartel[56] to prove b lowp in the supercritical case for a class

of solutions which are radiafly symmetric in some directions and have bounded variance in

others. ,iUternatively, Nawa [61/ shows a somewhat clifferent result (restricted to the case

ad = 2), namely that for al1 uo E H1(Rd) Nith H < O we have that either u(t) blows up in finite tirne, or u(t) is defrned for aIl t 2 O but that sup,,, - IIVuIIL2 = W.

For the vector case Iess is h o m . The virial theorem can be extended to the W L S

equation [32] and one obtains an identity simiIar to (3 .2) but nrith additional tenns. In the

case CI = 1 tliese t e m s drap out and we obtain finite-time blomp as in the scalar case.

The imposition of radial symmetry cannot be applied in the vector case, since the subspace

H:ad(Rd) of vector functions E for which El: . . . , Ed are radially symmecric is not invariant

under the flow generated by (1.33) urrless ûi = 1.

3.2 A blowup result for the scalar case

In this section we present a simplified version of the result OF Nawa [61], which asserts chat for

od = 2 and arbitrary data uo E lY1(IRd) and negative Hamiltonian we have either finitetirne

blomp or that the solution is defined for ali t 2 O but is unbounded in N'(Rd) . We also

generalize this result to the supercriticai case ed > 2. We begin by presenting a generalized version of the viriai theorem (3.2) valid for arbitrary weight functions.

Lemma 3.1

Let d? be a sufficiently smooth red weight function on IRd. Define the mriance by

Then we hâve

Proof

From (3.1) we have

and the first result follows. Similady. we get

giving the second result. The above caIculations are formal and assume that u is sufticiently

smooth, they can be made rigorous by the usual method of appro.ximating functions in

H ' ( R ~ ) by a sequence of smooth functions and then taking iirnits. I

The second resiilt of this lernma cm be viewed as the generaiization of the virial theorern

to arbitrary weight huictions. in the usual case O(x) = 1x1' we have

and (3.4) reduces to (3.2). Motivated by this resdt, we rewrite the Harniltonian in the form

Adding and subtracting 8H From the right hand side of (3.4) and using (3.6), we obtah

NOW, substitute (3.3) and (3.7) into the evolution equation for the variance

where

which is the generalized evolution equation for the variance for an arbitraxy weight function

@, No, we proceed to the main result of this section.

Assume r d 1 2, and let uo E H'(w~) have negative Hamiltoniarz. Let u be the solution to

(1.23) with initial condition uo. Shen either u blom up in HL(Rd) in finite tirne, or u exists

for di t 2 0 and rve have

Proof

The proof is a slighth sirnplified version of that given in [611. CVe proceed by contraàic-

tion. Assume that the resuit does not hoid. Then the solution exists for aIi t > O and there exists some LW > O such that

Define the viuiance V(t) as in Lemma 3.1, where the weight function d l be specified

shortly. Then clearly V ( t ) is non-negative for ai i t 2 O. We will derive a contradiction by

showkg that V ( t ) - -a as t - m. To do this, begin by defining 4 to be a reaI-valued h c t i o n in Cr[O, m) satisfying the

conditions

and set

where R is a positive constant which di be determined later in the proof. Inside the d-

cube [-R, RId we have Q(x) = l $ , while outside the larger d-cube [-ZR, 2 ~ 1 ~ we have @(x) = ZR? Note that supp (2d - A@) = fiR, where

QR = { ( x L , . . . ,Q) E IRd , ~ ~ Y ( ? C L ~ .. . ,Q) > R), (3.18)

is the exterior of the d-cube [-R, R ] ~ . In addition, we have

where C > O depends only on 4 and d. Referring to (3.9-3-14), we clearly have K ( t ) - 4Ht2 as t -, m. since the second term

in VI(t) is bounded in absolute value by some multiple of t. As weii, we have &(t) 5 O for

all t 2 O since ad > 2. Vie claim that, as long as R is chosen sufficiently large (depending on uo and LU), then we d i have &(t) 5 0, G(t) 5 IHlt2, and VL(t) 5 I ~ l t ' . It Nill then Follow that V(t) 5 2Ht2 for t sufficiently large, which wiil produce a contradiction. This daim nriii be established in the foUonring three lemmas. To simpMy the notation, from now

on C will denote a generic positive constant which may depend on a, d, and 4, but which is independent of -u and R.

Lemma 3.3

For aii t 2 O we hiive b(t) 5 0.

Pro of

From (3.16-3.17) it foiiows that &aj@ = O if i # j : while i f i = j we have aitlj@ = 6'' 5 2- Consequently, the integrand in the e'rpression for is always nonnegative, and the result

follows. I

Lemma 3.4

Assume that R is sufüciently large (depending on uo and M), and r e c d the set RR dehed

in (3.18). Then there e..sts F(uO1 LW) > O such that, for ally t 2 O, if

then rt-e have &(t) 5 1 Hlt2.

and hence, referring to (3.12) we have

and the result foilows by ttaking R2 $ E. From Lemmas 3.3-3.5, we see that in order to prove that V(t) -i -oo as t -i oo it is

sufficient to show that ~lu(t)ll&, 5 F(uO, LW). This is the subject of the ha1 lemma of the proof.

Lemma 3.6

Assume that R is sufficiently large (depending on uo and M.) Then for all t 2 O we have

with F(uo, hl) given in Lemmtl3.4.

Then we m n t to show that T = m. The method of proof is by contradiction, so assume

that T < m. We h o w From (3.14) and Lemmas 3.3 and 3.5 that

Also, for t E [O. Tl we have by the definition of T that ~[u(t) l l&~, 5 F(u0, LU) and hence

from Lemma 3.4 ive have G(T) 5 1 HIT'. Hence, referring to (3.9) and (3.10) we have

Xext, note from (3.16-3.19) that

IV@[- 5 Ca.

Hence, using the Cauchy-Schwartz inequality and (3.15) we can bound the second term as

a 3 ..

=Al* 1 'Y fu o eil w k Li

E $ I .a 8 El d

3 d fl

8 + H 9 + t3 O . d u

4 cd =i 0'

QJ

5' O'

3.3 Variance identities for the vector case

-4s previously mentioned, it is an open question as to whether negative energy solutions of

the n i L S equation blow up in 6nite time. A natural approach to this problem is to begin by

deriving a variance identity which corresponds to Lemrna 3.1 of the previous section. This

is done in the following lemma.

Let @ be a suEciently smoo th real weight Eunction on Rd. Defiae the variance by

Pi(t) = 3 LEj ( t ) ~ E, ( t ) ] , Qi(t) = B [ ~ ~ ( t ) a j E , o ] Furthermore, set ting

+ / (AB) ~ ~ ( t ) l ~ ~ + ~ , a i l

and

Pmof

Proceeding as in the scalar case, we have

= - 2 1 (ai@) (a3 [E,I),E;] + ( 1 - &)O [&ajE;l) and the &st result follows. Similady, we have

and

+ ( - a ) ( a i @ ) B [ a i E j a i a k ~ - ~ i a j a i a k E ; I 7 S establishing the second result. I

The expressions for Y and Z appear very similai., however tvhen we caiculate their t h e

derivatives we see an important ciifference. Each separates into three different components.

In the case of Y'? d of the components can be expressed as the gradient of a closed-form

e-xpression, while in the case of Zr this is only true for the first component. This is the crucial

problem which compiicates the expression for the variance in the vector case.

If we set @(x) = 1x1' this resuit reduces to the rnodified variance identity

similar to tliat obtained by Goldman and Nicholson [32]. For ad 2 2 and H < O this impiies that the quantity in brackets on the left side will eventuaily be negative. If a = 1 this is

simply the usual variance and and blonnip folioms as in the scdar case. However for ct # I this quantity is no longer necessarïly positive, and the proof breaks down. in this case the

problem remains open.

Chapter 4

Dynamic rescaling for single-peak solut ions

4.1 Self-similar solutions

In this section we consider an important class of blowp solutions of the W L S equation

which c m be mitten in esplicit form, namely the class of self-simiiar solutions. In the scalar

case such sohtions have been esensively anaiyzed by various authors, and we summârize

the most important results below. Most of these analyses have been carried out for the case

a = 1. wkch corresponds to a cubic nonlinearity and is of primaxy si,dcance physicdy.

Hence. in order to simpliSr the presentation, +ive MI1 assume cr = 1 for the remainder of this

chapter.

LVe are interestcd in constructing explicit solutions of (1.9) whi& blow up in finite t h e .

The simplest type of blomp solution is obtained in the critical case d = 2, where one can

use the pseudo-conformal transformation

1 i ~1x1' ,u(x- t ) - - e q ( ) u (A, ) , D - BC = 1 (4.1)

4 + Bt 4(A f Bt) -A+ Bt A, Bt to map globally defined solutions to blowup so1utions- hdeed, Cazenave [191 uses this corre-

spondence to anaiyze the asymptotic structure of critical blonrup solutions. If, in particdar,

we a p p l (4.1) nith A = t*, B = -17 C = 0, D = l / t*: and u given by the standing wave

solution u(r, t) = e.xp(zt)R(rj (where R is a ground state solution of ( M ) ) , we obtain a

solution of the form

However, this sohtion is unstable and has not been observed numerically [49].

A more general class of solutions can be obtained by first considering (1.9) in the radidy

çymmetric form

and then transforming it into new variables (p, [, T ) defined by

where L is a positive function OF time to be deterrnined later. Transformations of the type

(4.4) are usefui in studying blotvup solutions (see Section 6.1 of [71]) and are often referred

to as lens transformations, since they focus the new coordinates near the origin. Applying

this transformation to (4.3) we obtain

where a(t ) = -LLt. For self-similar solutions we require a(t) to be a positive constant, which

we denote simply as a. This leads, at least formally, to a Camiiy of solutions of the fora

where Q : [O, oo) - @ satisties the boundary value problem

which has the form of a non-Iinear eigenvalue problem with a as the eigenvdue. Yote thae

the dimension d cm be considered to be a real-valued parameter. The existence of solutions

for (4.7) has beea studied by several authors. It vas shom by Wang [76] for the case

d = 3, and Iater by Budd, Chen, and Russell [13] for the more general case 2 < d < 4 that for aIl Q(0) E B and dl a E R there exists a unique solution on [O, ou) satisfying the

second boundary condition Q(m) = O. However, if we require solutions of (4.3) to have

finite enerm, then an additional constraint must be placed on the parameter a. Indeecl, as

shom by LeMesurier, Papanicolaou, Sulem and SuIem [49], any solution of (4.7) s a t i s w g

IIVQII L2 < cwi and IIQIILd < oc must necess* have infinite ~"norm, and furthemore its Hamiltonian must mnish,

AS pointed out in [481, this suggests that there is at most a discrete set of values of a for

which dowable solutions exist.

Indeed, in a series of numerical simulations, Budd, Chen, and Russell [131 showed that for

each d > -2 there is a coimtable inhity of pairs (Q, a) for which a solution to (4.7) satisfying (4.5) exists. These pairs Lie on branches which, as d -t 2, bifurcate h m either the ground

mate R or the zero function. In other words, as d -t 2, we have either (Q, a) -3 (R,O) or

(Q, a) 7 (0, O). Yumericd simulations suggest that only one of these branches, namely the

branch comprising the solutions constructed in [491, corresponds to stable solutions of (4.3).

Functions Q lying on this branch are characterized by the fact that they are monotone for

values of d close to 2. The other branches are characterized by the presence of multiple

bumps. Further analysis of these muiti-bitmp solutions tvas carried out by Budd [MI.

For the critical case d = 2 the situation is quite different. As noted above, the constraint

(4.5) forces a to vanish as d tends to 2, which suggests that there are no critical solutions

of the form (4.6). Indeed, one can compute the asymptotic expansion of Q at inhi ty as in

Section 3.1 of [48]. and an analysis of this expansion shows that there are no solutions of

(4.7) which satisfy the boundary condition Q(m) = O if d = 2 and a > O. This problem was analyzed by Landman, Papanicolaou, Suiem and Sulem [46] and LeMestrier, PapanicoIaou.

Sulem and Sdem [43], who constructed formd solutions for the case d = 2 as the asymptotic

limit as d - 2 of solutions of the form (4.6). These solutions are gîven by

where r is d e h e d in (4.4), and L(t) has the decay rate of the supercritical case modified by

a slowly changing correction term,

As a consequence, the quantity a(t) = -LLt no longer approaches a positive constant, but

instead decays extremely slowly to O at a rate &en asymptoticdy by

Il- - log r + 3 log log r - (4.11) En the lirnit r - cc we have a 3 O, and Q(+, a(t)) 4 R(-), so that (4.9) c m be repIaced by

The importance of the self-similar solutions constructed above lies in the fact that they

appear to represent the limiting asymptotic behavior for ail solutions which blow up at a sin-

gle point. For the supercritical case, direct numerical simulations of isotropic solutions were

performed by Budueva, Zakharov and Synakh 1151 and Goldman and Nicholson [33], r u e

LeMesurier, Papanicolaou, Sdem and Sdem [4Q] and McLaughlin, Papanicolaou, Sulem

and Sulem 1571 studied blomp using the method of dynamic rescaling. Anisotropic solu-

tions were also analyzed using dynamic rescding in [47]. Mternatively, Alaivis, Dougah

and Karakashian [21 and Akrivis, Dougalis, Karakashian and McKinney [3] deveIoped an

extremely accurate method for simulating radially symmetric solutions of the NLS equation

using an adaptive Garerkin finite element method. In aii cases? the asymptotic fonn of the

solution mas observed to be

up to transiations and phase changes. The quantity a was observed to be independent of the

initial conditions, and in the case d = 3 its d u e was computed to be approiamately 0.9174.

In the critical case, experimentd verification of the predicted bIowup behavior is Car

more difficult since the correction term for the blomp rate varies very slowly, and since

the asymptotic regime is only attained extremely close to the blowup tirne. The basic form

of the solution (4.6) w u verified using dynamic rescaling in both the isotropic [49] and

anisotropic [47] casest while the more delicate problem of verifying the Log-log decay law

(4.10) ivas analyzed in [3]. These analyses aU concluded that (4.9) indeed represents the

Limiting behavior of single-point blomp solutions in the critical case.

Much of this analysis above can be carrïed over to the vector case. Indeed, trmsforming

(1.32) into new variables (E.

and a = -LLt is an arbitrary positive constant. Notice that in the special case a = O

this reduces to the bound state equation (2.2). As in the scalar case, admissible solutions

can exist only for a certain set of values of a, which depends on the dimension d and the

parameter a. Indeed! we have the followïng result.

Lemma 4.1

If d f 4 and Q is a solirtion of (4.16) for which \IVQllr;2 < ~1 and IIQII L4 < cm, then ive have llQ1lLz = cm and

Proof

The proof is essentidy the same as that for the scalar case (see Section 7.1 of [71]).

Mdtiplying (4.16) by aA$+ (1 - a)V(V. O), taking imaginary parts, and integrating, -ive obtain

Applying integration by parts twice, the fist integral becomes

For the second integral, we replace dq+ (1 - a )V(V .Q) by o + i a ( x V ) q - I Q ~ ~ Q and i n t e p t e by parts to obtajn

- a(d - ') j 1 ~ 1 1 ~ . 4

Substituting the last two resrtlts into the first equation @es H(Q) = O. To show that the

mass k infinite; suppose that Q E L'(Rd), multiply (4.16) by q, take the imaginary part, and integate over Rd to obtain

Since d > 2 it foilows that Q = O. Consequently, for non-trivial solutions Q we cannot have Q E L'>(R~). I

The numerical construction of solutions of (4.16) appears to be fairly difticult. hdeed, the

methods used in Section 2.5 to construct ground state solutions R are no longer applicable,

since the operator obtained by linearizing the left side of (4.16) is no longer self-adjoint, and

the conjugate gradient met hod cannot be applied. There exïst a number of generalizations of

this method which are applicable to non self-adjoint operators. However, the implernentation

of these methods to the problem in question was not successfui, indeed the quasi-linearization

process faiied in most cases to converge to a solution. This appears to be a difficult numericd

problem.

4.2 The method of dynamic rescaling

In order to study the asymptotic structure of blomxp solutions numericdy, it is necessq

to use a method which evolves dynarnically so that it captures the structure of the solution

near the bloivup point. One such method which has been applied with considerable success is

d-vnarnic rescahg. This method was first developed for the radially symmetric NLS equation

by SIcLaughlin, Papanicolaou, Sulem and Sulem [571 and LeMesurier, Papanicolaou, SuIem

and SuIern [491, and then extended to the anisotropic case by Landman, Papanicolaou, Sdem

and Sdem [47]. It was also used to simulate blowup solutions of the W L S equation and

the Zakharov system by Papanicolaou, Sulem, Sulem and Wang [64].

Here, we present a summary of this method as it is appiied to the cubic W L S equation.

Blotivated by the lem transformation used in the previous section, we define a new set of

variables (E ,

and L ( t ) is given by

-= LT- L2(t) t= . L A, ( t) ' After some computations [64] we find that E, Ai, xo, and O satisfy the foiionring system of

equat ions,

In (4.33-4.25) the scalar and tensor differential operators A:, and are given by

where VV denotes the matrix difFerentia1 operator whose elements are @en by

and the scdar prodiict is defined by

Mso, G = (gi j ) and il = (a,) are d-by-d matrices and f is a vector in Eld, given respectively

where B = (bij) is an d-by-d matrk and P = (pi) is a vector whose elements are given by

b.. - a.. t a - t a ,

Xotice that the equations for E and X i decouple from the equations for x,-, and 0.

ive Mplement this method numericalIy using a finite ciifference method. FVe work on

the cube [-L, LId for some su£Eciently large LI and divide the interval [O, L] into n equai

subdivisions. This gives a total of (2n + l )d points where E must be caicuiated. ÇVe discretize the spatial differential operators using finite ciifferences. Since the structure of the sohtion

in rescaied coordinates remains relatively constant after the initiai transients have died off.

ive found that the simplest method of tirne discretization was to use a classicd fourth order

Runge-Kutta scheme with fked timestep. The evaluation of the spatial integrals is done

using Simpson's method.

For boundary conditions, ive approximate the solution by extrapolating to fictitious points

outside the domain of integration. This corresponds rnathematicaiiy to neglecting the effect

of d u e s of E outside the region of integration at previous times when calcuhting its mlue

at b o u n d q points. This approximation is accurate for blonrup solutions. Indeed, near the

bottndary of the region & is srnaII in magnitude, and consequently we c m neglect the cubic

term in (4.22). The resuiting equation is h e a r and c m be solved exactlÿ using the method of

characteristics. .h analysis of these characteristics [47] shows that they tend to move away

From the origin as time increases so long as a11 of the Xi's are decreasing in time, which is

true for blowup solutions af'ter an initial transient. This approximation c m also be justified

numericaiiy by perforrning the integation on a Iarger region (with the same mesh spacing)

and veriFying that sirnilar resuits are obtained.

4.3 Single-point blowup in two dimensions

The lïrst case we consider is the critical case corresponding to d = 2 and a = 1. For o u

simdations we used parameters L = 30, n = 50, and p = 3, and we discretized the spatial differential operators using a 7-point scheme in each direction,

bVe performed a series of simulations with four d8erent values of a, nameiy 0.5, 1.0, 2.0,

and 5.0, and with initial condition

for X = 1, X = 3, and X = 3. We were able to continue the simulations up to values of r on

the order of a thousand, at which t h e the amplitude of the solution in physical variabIes

[vas greater than 10'" By this point the solution had stabilized to the form

, ) = ( T ) Ê(o, T ) = (c, O), c > O, (4.35)

where the profile & varied elrtrerndy dowly. The an,pular frequency of E at the origin, which

can be compiited as

did not approach a hmiting constant but rather conthued to vary extremely slowly, which

is consistem Mth a nearly self-similar behavior. Using (4.21) and (4.23), we cornpute the

b 1 0 ~ p rate in tems of aji? Xi and w as

where we have divided by ur in ordcr to obtain a physicdiy invariant quantity. As expected,

a ( r ) ivas observed to decay extremely slowly to zero. A naturai question is to determine

whether a blomp rate of the form (4.10) holds as in the scalar case, which is equivalent to

studying the decay rate of a(t ) . In analogy Nith (4.11): we hypothesize for the vector case a decay rate of the form

X (4.38) a'T) hi

T f 3 10g log T'

with X depending on o. To determine the validity of (4.38) ive performed a series of sirnu-

lations with initial condition Eo = 6e?cp(-x? - x:) using parameters L = 30 and n = 75 for increased accuracy. In Emme 4.1 nre have plotted the quantity

as a function o f r for a = 1.

Tt appears that X(T) is converging extreme1y slowIy to some value which is fairly near the

expected ratio X = sr. To make this more precise, we fit this cuve to the function

where the parameters A, B and C are chosen to minimize the quantity

Fiowe 4.1: Evolution of quantitÿ X(r) = a( r ) (logr + 3 log logr) for a = 1.

with ri a sequence of n values which tend to r,,. The number of points n was taken to be

50 for al1 calculations. The form of the function aven in (4.40) was chosen because it gave

reasonable values for the parameters A, B, and C, and was relatively insensitive to the value

of Tm,.

The resiilts which we obtained for several different values of a are shown in Fi,we 4.2.

For a = 1 we compute a limiting value of A = 3.19, which is in good agreement with the

theoretically predicted value A = ri h: 3.14. The error term x was quite smaii relative to the d u e a = 0.24 at the end of the simulationt indicating that X,,,,,(r) is a dose fit to X(T).

For other values of a we observed siightly larger errors, but again the fit was quite good.

The limiting value X appears to be a decreasing function of the parameter a.

Fiove 4.2: Determination of limiting value of X(r) for various values of a.

We now tum to the study of the IMiting profile E. Using (4.17) to convert hom rescaled to physical coordinates, and eliminating the angular frequency w using a spacetime dilation

(1.41), we find that (4.35) corresponds to a solution of the form

where

and the stretching factors -il and -fi are @en by

W ) -/? = iim - .-.. L ( T )

As expected. the limiting values of yi and were independent of Eo; we List the d u e s

obtained in al1 four simuiations in Fi,we 4.3 below.

I 1 1 1

Figure 4.3: Limiting d u e s of 71 and as a h c t i o n of a!

X comp~arison of (4.42) and (4.9) suggests that A should approach the ground state R

corresponding to the value of a! used in the simulation, To test this hypothesis, nte tabulate

in Figure 4.4 the L' and Lm-norms oE A and R, which agree to within a few percent, where

the data for the ground states have been computed Erom the numerical results obtained in

Section 2.4. Siniilarly, in Figures 4.0-4.8 RE plot the level curves of ]Al and IR! for each value of a!, using ten e q u d y spaced contours. Xote that for a! = 0.5 the contours are rotated

by an angle of r/2, This is to be expected, since the initial condition Eo is elongated in the

x2-direction, while ground states corresponding to a < 1 are elongated in the xI-direction.

4.4 Single-point blowup in three dimensions

In this section we consider the resuits obtained in the supercritical case when d = 3 and

o = 1. For al1 of the simuiations we used parameters L = 10, n

discretized the spatiai differential operators using a 5-point scheme.

= 30, and p = 3, and

Fiogre 4.4: Cornparison of L%d Lm-noms of A and R for various values of a.

We began by performing a series of simulations with four different values of a, namely

a = 0.5, o = 1.0, a = 2.0, and a = 5.0. For each of these values we used three different

initiai conditions, @en by

with choices ( A l p ) = ( 1 , l), (A, p ) = (1,2)? and (A , p ) = (1,3).

As espected, the dynamics were much more rapid than in the critical case, and it

only necessq to continue the simulations up to values of T of several dozen. By this time,

the rescaled solution E had stabilized to the form

where the limiting angular fiequency w can be cornputed explicitly as

LJ = lim

in Fiogre 4.9 we tabulate the value of w observed in each case.

Yotice that no value is given for ai = 0.5 and (A, p ) = (1,3). This reflects the fact that

in this case the simulation broke down due to the onset of splitting, a phenornenon in which

an initidy single-humped profile divides into two separate peaks as its amplitude increases.

The dparnic rescaling method in its current form is unable to mode1 the l i m i t a behavior

of such solutions, they \di be investigated Further in Chapter 6.

Using (4.21) and (4.23), we c m compute the lirniting vaIue of the blowup rate a in te-

of aii, Ai and w as a = -

where we have normalized by dividing out the factor w to obtain a

quan t i . The corresponding values obtained for each simulation are

Fiove 4.5: Contour plots OF [Al (left) and IR1 (right) for (a, d) = (1,2) and ai = 0.5.

Fi,oure 4.6: Contour plots of IAl (left) and IR1 (right) for (a, d) = (1,2) and a = 2.0.

Fieme 4.7: Contour plots of IAl (left) and \RI (right) for (n! d) = (1.2) and cr = 2.0.

Figure 4.8: Contour plots of IAl (left) and [RI (right) for (a, d) = (1,2) and a = 5.0.

1 (Al P) = (1,l) 1 (A, CL) = (1,2) 1 (A, p ) = (1,3) a! = 0.5 1 0.301677 1 0.301674 1

Fi,we 4.9: Lirniting d u e of an,dar frequency w as a function of cr and Eo.

Note that the blowup rate is independent of the initiai condition, and that for a = 1 its

talue is appro'rimately 0.9174, in excellent agreement with the value obtained in the scaiar

case (see for example Section 3 of [131).

Fi,we 4.10: Limiting d u e of blonrup rate a as a hnction of û. and initial condition Eo.

To determine the nature of the dependence of the blotvup rate on the parameter a? ive

ran a series of simulations for a large range of values of CI using an initial condition EO

corresponding to (A. p ) = (1, l), and caiculated a in each case using (4.48). In figure 4. I l we

plot the observed value of a against a. Notice that a attains its maximum d u e at ar = 1.

Yen- ive turn to the anaiysis of the rescaied profile E. Referring to (4.17), we rewrite

(4.46) in terms of the physical coordinates x, t , and E as

where the rescaled profile A is d e h e d by

and where -/l1 72, n/3 are stretching factors given by

respect iveIy-

.A cornparison of (4.49) with the corresponding scalar law (4.13) suggests that A should

identify with a solution Q of (4.16), with a given by the observed rate computed in (4.48).

As well, we e-qect from Lemma 4.1 that A shodd have zero energy. The fact that A solves

(4.16) is eaçy to verify, indeed it follonrs aimost imrnediately From substituting (4.46) into

(4.22) and using the fact that G = 0: f = 0, and L-'L, = -da. However, the verifkation

that H(A) = O cannot be cacried out with this method: since we know the d u e of A o d y

in a region near the origin. In the scaIar case, Budd, Chen, and Russell [13] are able t o

constn~ct Q nurnericaily and b d values of a for which H ( Q ) = 0, however this method

reiies on the radial symmetry of Q and hence does not appear to be applicable in the vector

case.

ive conchde this section by presenting a series of contour plots of [Al for various values

of a in Figures 4.12-4.14. As is the case with the gound states R constructed in Section

2.5, the contours of IAI are rotationally symmetric in the X Z - X ~ pIane and eilipticai in form

in any pIace containing the xL-axis. To &sIay these contours we show two sets of plots.

On the Ieft we show a transverse view (a cross-sectionai view in the 22-53 plane), while on

the right we show a longitudinal view (a cross-sectionai view in the XI-xz or XI-x3 plane).

For a = 1 we find as expected that A is sphericdy symmetric. As a increases A becomes

more oblate in the x? and x3-directions, mhile as a decreases A becomes more prolate in the

xL-direction. This behavior is similar to that which was observed in Section 2.5.

-4 -2 O 2 4 6

(a) Transverse view

-4 -2 O 2 4 (b) Longitudinal view

Figure 4.12: Contour plots of rescaled profiIe [Al for (a,d) = (1,3) and û = 1.0.

_I

-6 -4 -2 O 2 4 6

(a) Transverse view

J - 6 - 4 1 2 0 2 4 6

(b) Longitudinal view

Fi,we 4.13: Contour plots of rescded profiIe \Al for (c7 d) = (1,3) and a = 2.0.

-1 2 -8 4 O 4 a 12

(a) Trawerse view

-1 2 -8 -4 O 4 8

(b) Longitudinai view

Fiogre 4-14: Contoirr pIots of rescaled profile IAl for (r, d ) = (1,3) and a = 5.0.

Chapter 5

A dynamic mesh refinement technique for singular solut ions

5.1 Multi-peak solutions

In the previous chapter, the method of dynamic rescaling was seen to be highly effective

for determining the asymptotic structure of singlepoint b1owup solutions of the NLS and

V'iLS equations in the neighbourhood of the blonnip point. In some situationst however,

nie are aiso interested in the structure of the solution at points which are further away hom

the singularity. This is especially the case if ive are studying solutions with more than one

blowup point.

When dynamic rescahg is applied to a multi-peak problem, such as simulations of the

WLS equation for smaii values of the parameter a? the method quickly breaks dom. The

reason for this is as foiiows. Suppose for simplicity that we are in dimension two and that

bloivup occurs at the points (O. c) and (0: -c) on the x2-axis. As blowup progresses, the

quantities A l and X2 Nill evolve so that (4.20) is satisfied. Under this evolution, we di

have A l - O as before, but X2 dl be bounded away from