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BAMC 2012 UCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 March 2012

Emergence of dark solitary waves fromlarge-amplitude spatially-periodic waves

Thomas J. Bridges, University of Surrey

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Classical NLS dark solitary waves

Defocussing NLS,

iAt + Axx − |A|2A = 0 .

has an exact DSW solution

A(x , t) =√

2 ei(kx−t)(k + iβ tanh(βx)),

with β2 = 12(1− 3k2).

Bi-asymptotic to a periodic state with a phase shift

A(x , t)→√

2 ei(kx−t)(k ± iβ)

as x → ±∞ .

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Mechanism for emergence of DSWs

What is the mechanism for the emergence of DSWs?

Can we generalise the creation of DSWs to non-integrablesystems?

Chicken and egg game: what comes first? state at infinity,connecting orbit?

Argument: take an arbitrary periodic pattern, determineconditions for the emergence of a dark solitary wave.

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Spatially periodic states of NLS

iAt + Axx + A− |A|2A = 0 .

Spatially periodic states

A(x) = A0 eikx with k2 + |A0|2 = 1 .

k

|A |0

Transition from (spatially) elliptic to hyperbolic at k2 = 13 .

T.J. Bridges (Surrey) Emergence of dark solitary waves

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KIVSHAR (1990) Physical Review A

Kivshar notes that as k2 → 13 , the DSW goes to a sech2 solitary

wave. He proposes that the birth of a DSW (a “low amplitudeDSW”) is governed by the KdV equation.He proposes a solution of the form

A(x , t) = (A0 + B(X ,T ))ei(kx+φ(X ,T )) ,

whereX = εx and T = ε3t ,

and shows that B(X ,T ) satisfies (to leading order) a KdVequation

c0BT + c1BBX + c2BXXX = 0 .

– see also SAUT ET AL (2009), CHIRON & ROUSSET (2010)

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Features and generalisations

NLS→ KdV: integrable PDE→ integrable PDEKIVSHAR, ANDERSON & LISAK (1993) Physica Scripta:showed a similar bifurcation for the non-integrable NLS

iAt + Axx + f (|A|2)A = 0 .

How can this be generalised?

The change along the periodic orbit is a saddle-centretransition of a periodic orbit of a Hamiltonian system.Can also be interpreted as a collision between two periodicorbits, one elliptic and one hyperbolic.

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(Spatial) energy-wavenumber space

The NLS is Hamiltonian

iAt =δHδA

with H = |Ax |2 − |A|2 + 12 |A|

4 .

However, it is the spatial energy that is of importance along thebranch of spatially-periodic solutions

S = |Ax |2 + |A|2 − 12 |A|

4 .

Evaluated on the branch of periodic solutions

S(k) = 12(1− k2)(1 + 3k2) .

k

S(k)

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Collision of two periodic orbits

k

S(k)

The elliptic periodic orbit has a Krein signature.

This scenario that can be found in general in Hamiltoniansystems.

Jux = ∇S(u) , u ∈ R2n , n ≥ 2 .

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Embed in a PDE with Hamiltonian spatial part

Consider PDEs of the form

Mut+Jux = ∇S(u) , u ∈ R2n , n ≥ 2 .

That is PDEs with Hamiltonian spatial part, where M is at thispoint an arbitrary constant 2n × 2n matrix.

Suppose the steady system has a collision of periodic orbitsassociated with an elliptic-hyperbolic transition. Then nearbydynamics is governed (to leading order) by

m′(k)BT + S′′(k)BBX + K BXXX = 0 .

A KdV equation with geometrically determined coefficients.

T.J. Bridges (Surrey) Emergence of dark solitary waves

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KdV→ DSWs

m′(k)BT + S′′(k)BBX + K BXXX = 0 .

Here X = εx , T = ε3t , m(k) is the momentum, and S(k) is themomentum flux. K is the Krein signature of the elliptic periodicorbit in the collision.

The coefficient of the nonlinear term is determined by thecurvature of the momentum flux (spatial energy) as afunction of wavenumber.The sign of the dispersion is determined by the Kreinsignature of the collision.

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Strategy

Given a PDE, with the steady part a Hamiltonian system

Jux = ∇S(u) , u ∈ R2n , n ≥ 2 ,

plot families of periodic orbits in the energy-frequency(momentumflux-wavenumber plane in the spatial case) andidentify points where

S′(k) = 0 .

Such points are indications of a saddle centre transition.Extend to the time-dependent case. The nearby dynamics isgoverned, to leading order, by

m′(k)BT + S′′(k)BBX + K BXXX = 0 .

How do we show this?

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Saddle centre transition

At the transition, +1 is a (spatial) Floquet multiplier of geometricmultiplicity one and algebraic multiplicity four.

Jordan chain (θ = kx + θ0)

Lξ1(θ) = 0 , Lξ2(θ) = ξ1(θ) , Lξ3(θ) = ξ2(θ) , Lξ4(θ) = ξ3(θ) .

How to scale the Jordan chain?Propose solution of PDE of the form

u =(εξ1)φ+(ε2ξ2

)B+K

(ε3ξ3

)v−K

(ε4ξ4

)I+ε5W (X ,T , θ, ε) .

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Modulation equation

Project onto the generalised eigenspace,

φX = B + · · ·

BX = K v + · · ·

m1φT − vX = I − 12κB2 + · · ·

−m1BT − IX = 0 + · · · ,

with

κ = −2〈〈ξ2,N (ξ2, ξ2)〉〉 and m1 = 〈〈ξ2,Mξ1〉〉 .

Can prove that κ is proportional to S′′(k) (a few pages!).Combining the above equations gives the KdV equation forB(X ,T ), with the phase shift incorporated in φ(X ,T ).

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Four cases

m BT + κBBX + e± BXXX = 0

κ > 0 κ > 0

κ < 0 κ < 0

ω ω

II

I

ω ω

I

he+ e−h

h e+ e− h

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Example – KdV planforms

Consider the elliptic PDE (a variant of the steadySwift-Hohenberg equation)

wxxxx − Pwxx − wyy + w − w2 = 0 .

The y−independent part is Hamiltonian with (spatial energy)

S = 12w2

xx + 12Pw2

x − 12w2 + 1

3w3 − wxwxxx .

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Steady SH equation

The steady SH equation

wxxxx − Pwxx − wyy + w − w2 = 0 .

can be written in the form

Muy + Jux = ∇S(u) , u ∈ R4 ,

with

M =

0 −1 0 01 0 0 00 0 0 00 0 0 0

.

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KdV planforms

At transition points, where S′(k) = 0, the theory goes throughand the modulation equation is KdV – in the (x , y) plane

m′(k)BY + S′′(k)BBX + K BXXX = 0 ,

with coordinates

X = εx and Y = ε3y .

KdV has a whole range of N−soliton solutions which becomeN−soliton planforms in this case.

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Two-soliton solution of KdV

The KdV equation has n-soliton solutions. In this case theygenerate n−soliton planforms. Example: 2-soliton

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Water waves

Look at the steady water wave problem along branches ofspatially periodic waves – with c fixed and the wavenumbervarying along a branch. Does a saddle-centre transition ofFloquet multipliers occur? Yes!

In infinite depth: BAESENS & MACKAY (1992) J Fluid MechIn finite depth: VANDEN-BROECK (1983) Physics of FluidsAt low amplitude coupled to mean flow: TJB &DONALDSON (2006) J Fluid Mech.

The theory proposed here suggests that near each of thesepoints, the appropriate model is the KdV equation.In the first case, the KdV is a model equation for water waves ininfinite depth!

T.J. Bridges (Surrey) Emergence of dark solitary waves

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DSWs in the water wave problem

The appearance of DSWs in the water wave problem is not sosurprising in finite depth since the NLS model in finite depth isthe de-focussing NLS, which has unsteady DSW solutions.

To find these DSWs at finite amplitude, look for saddle centretransitions of spatial Floquet multipliers.

Invoking the above theory then indicates that the KdV equationis the nonlinear model nearby.

New mechanism for the appearance of KdV, and it shows thatKdV can also be model for water waves in infinite depth.

T.J. Bridges (Surrey) Emergence of dark solitary waves

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Mechanism for KdV?

Shallow water is not necessary for the appearance of KdV as amodel equation for water waves.

Indeed, “shallow water” is neither necessary nor sufficient forthe appearance of KdV as a model equation for water waves.

A new mechanism for the appearance of the KdV equation as amodulation equation.

Other mechanismsA dispersion relation determines whether KdV is theappropriate modulation equation.“Nonlinearity balances dispersion” implies the appearanceof KdV as a modulation equation.

T.J. Bridges (Surrey) Emergence of dark solitary waves