On the Toda shock problem

Physica D 65 (1993) 242-266 North-HoUand

SDI: 0167-2789(92)00022-I

O n t h e T o d a s h o c k p r o b l e m

S p y r i d o n K a m v i s s i s 1

Courant Institute, New York, USA

Received 20 December 1991 Revised manuscript received 10 November 1992 Accepted 18 November 1992 Communicated by A.C. Newell

We consider the long time behavior of the (doubly infinite) Toda lattice under shock initial data, for the critical and subcritical case. Our method consists of analysing the Dyson formula for the solution. The integrals in this formula are of Laplace type and their behavior is extracted by solving a variational problem, in the spirit of the Lax-Levermore- Venakides method. The final answer is obtained by identification with a particular (modulated) algebro-geometric solution of the Toda lattice. We also consider the long-time behaviour of the Toda lattice under a local perturbation of the initial data in the supercritical case. Our asymptotics are identified with a degenerate algebro-geometric solution. Physically, we see solitons in a periodic background.

1. Introduction

1.1. Statement and background of the problem

In this p a p e r we cons ide r the T o d a la t t ice of

i n t e r ac t ing par t ic les . Physica l ly , this is a one-

d i m e n s i o n a l la t t ice of par t ic les in te rac t ing with

e x p o n e n t i a l forces ,

~ . = y n , (1 .1.1a)

)~. = e x p ( x . _ l - x . ) - e x p ( x . - x . + l ) • (1 .1 .1b)

H e r e x . is the d i s t ance o f the n th par t i c le f rom

the or ig in , a n d y . is i ts ve loci ty . T h e do t deno t e s

d i f f e r en t i a t i on wi th r e spec t to t ime . AS is well

k n o w n ([5]; see also [11, 13]), these equa t ions

a re H a m i l t o n i a n , ( fo rma l ly ) c o m p l e t e l y integ-

r a b l e , wi th a L a x - p a i r r e p r e s e n t a t i o n

L = [ L , B ( L ) ] , (1 .1 .2)

Correspondence to: S. Kamvissis, Ddpartement de Mathd- matiques, lnstitut Galil6e, Universit6 Paris-Nord, Av. J.-B. Cl6ment, 93 430 Viiletaneuse, France. E-mail address: [email protected].

w h e r e L and B(L) are the doub ly infini te tri-

d i a g o n a l ma t r i ce s (" ) a _ l b _ l

L = b_ 1 a o b 0

b0 a l bl

and B(L ) =

i 0 b_ 1

- b 1 0 b o

- b 0 0 b 1

T h e va r i ab l e s a. and b. are c o n n e c t e d to the

ini t ia l va r i ab l e s x . and y . as fol lows:

a,, = - ½y, , , bn = 1 exp[½(x. -x .÷ l ) ] . (1.1.3)

Wri t ing ou t (1 .1 .2) expl ic i t ly , we have:

0167-2789/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved

S. Kamvissis / On the Toda shock problem 243

ti. = 2(b2. - b2_a), (1.1.4a)

lJ,, = b, ,(a.+, - a . ) . (1.1.4b)

In particular we consider equations (1.1.4) under initial conditions corresponding to the so-called Toda shock problem:

Suppose we have a distinguished particle driven at fixed velocity 2a (where a > 0) into a semi-infinite Toda lattice. Numerical experi- ments ([6,7]) indicate that in the moving frame, the particles eventually settle down into a purely time-periodic motion (which is also asymptotically spatially periodic of period 2), provided a > 1, while they eventually come to rest if a < 1. Our numerical calculations show that also in the case a = 1 the particles eventually come to rest in the moving frame. Recently, Venakides, Deift and Oba have given a complete derivation of the results of ref. [6] for the case a > 1 in ref. [13]. This paper contains the analysis of the long time behavior of the Toda lattice in the cases a = 1 and 0 < a < 1, as well as the analysis of the case a > 1 with initial conditions which are a local perturbation of the original Toda shock problem. Our results are described in section 2 below.

Doubling up the system, the above (non- autonomous) Toda shock problem reduces to an initial value problem for the autonomous doubly infinite Toda lattice, i.e. equations (1.1.4) above, with n E E, satisfying the symmetry conditions

a , ( t ) = - a _ , ( t ) , b , ( t ) = b_ ,_x ( t ) (1.1.5)

and with initial conditions:

ao(O ) = O, (1.1.6a)

a.(O) = sgn(n) • a for n ~ O, (1.1.6b)

b.(O) = ½ for n E Z . (1.1.6c)

(Note that the conditions (1.1.5) follow from (1.1.4) and (1.1.6).)

In terms of x, and y , the Toda shock problem is expressed by equations (1.1.1), for n E Z, together with

x, = O, (1.1.7a)

y , = - sgn(n) - 2a for n # 0 , (1.1.7b)

Y0 = 0 . (1.1.7c)

Note that it follows from the above that

x 0 = 0 for all t . (1.1.8)

The reason for the criticality of a = 1 is the following: in the case a > 1 the spectrum of L (preserved by the flow) consists of two bands [ - a - 1, - a + 1] and [a - 1, a + 1] together with an eigenvalue at zero, while in the cases a < 1 and a = 1, the (continuous) spectrum of L consists of one band, the interval [ - a - 1, a + 1].

Remark . The initial condition b , ( 0 ) = 1 implies that all the particles are initially placed at the same position. It is easy to transform the problem with b,(0) = b0 for any b 0 into the one with b,(O) = 21-.

1.2. Solution o f the prob lem

The basic model for the work that follows is actually [13] following methodologies developed in refs. [9] and [14].

(i) The case a = 1. In this case, the Toda lattice eventually comes to rest, with the a , ' s approaching zero and the b , ' s approaching the limiting value 1. Thus, under the Toda dynamics the initial lattice configuration is transformed eventually into a different configuration as the time goes to infinity. In fact the complete picture for large times is the following: Define N =- n/ t . For large values of N, both a , and b, - 2 t- are exponentially small: the shock is not yet felt. There is a critical value N* for N (giving the shock front speed) at which the shock first appears. For N < N*, the lattice undergoes decay-

244 S. Kamvissis / On the Toda shock problem

ing oscillations and an explicit formula describing these oscillations is derived. In particular, the rate of decay is of order 1/t.

The proof proceeds as follows: first one de- velops the (direct and inverse) scattering theory for operators with nonstandard asymptotics,

an-">l as n - - ~ ,

an"">--1 a s n - - - > - ~ .

The solution of the inverse problem is obtained via a Marchenko type equation. Then the above Marchenko equation is solved to obtain an expression for the solutions of.the flow an(t ) and bn(t), involving Fredholm determinants. Finally, after some manipulations one obtains a Barg- mann-Dyson-type formula (el. [4]) for b, as a ratio of infinite series of multiple integrals of real exponentials. These integrals are evaluated asymptotically by solving a maximization problem, with a certain quantum condition, as con- straint, in analogy with refs. [14] and [13]. Sub- stituting the maximizers into the multiple integrals, we get an expression for the infinite series involving theta functions, which one recognizes as the solution of the 1-gap quasiperiodic (in space) Toda lattice. The moduli of these theta functions vary (slowly) with N and in fact as N approaches zero, the gap closes. Making use of identities involving theta functions we obtain an explicit and simple formula for the answer, when N is small.

This settles the case where N < N*. For N > N*, the answer follows trivially from the Dyson formula. Also, an explicit formula for the shock front velocity is derived.

(ii) The case a < 1. In this case the lattice also eventually comes to rest. The limiting value of the an's is zero, while that of the bn's is ½ (1 + a). One distinguishes three different regions:

There is a region (as for a = 1) where the shock is not felt yet (N > N*). For intermediate N, the lattice undergoes decaying oscillations. Finally, for small N the oscillations are very small and the lattice settles to rest.

The analysis of this case is similar to the case above, at least in the first two regions. When N is small, the problem is more subtle because the "quantum condition" which usually enables us to capture the oscillations, is ineffective here. In this case, our method computes the limiting values of the an's and bn's to leading order, but without the oscillations.

(iii) The generalized Toda shock problem. Here one deals with initial data L(0) that are a compact perturbation of the data (1.1.6), with a > l .

In this case the spectrum of L will consist of the two bands [ - a - 1, - a + 1] and [ a - 1, a + 1] as well as several (say p) bound states. It turns out that only the ones that are in the gap [ - a + 1, a - 1] play a significant role in the asymptotic behavior of the lattice (for small enough N).

The result is the following: for small enough N, after the shock front, the solution can be identified as the degenerate (p + 1)-phase algebro-geometric solution with the following structure: there are p + 1 gaps 'separated' by the bound states. It can be written down explicitly in terms of simple expressions (at least when p is small). One interprets this solution as follows: each bound state corresponds to a soliton in the background of a period 2 lattice. For non-zero bound states we see travelling solitons with speed depending on the energy of the bound state, a phenomenon which is also readily ob- served numerically. For a zero bound state we see a "breather", as in ref. [13]. We also observe a phase shift in the lattice between the left and right sides of the travelling solitons.

Note that the cases a = 1 and a < 1 are trivial, since the continuous spectrum of L consists only of one band, so there is no gap at all.

1.3. The Dyson formula and the variational problem (for both 0 < a < l and a = 1)

From now on we only consider non-negative n. The results for negative n follow immediately from the symmetry relations (1.1.5).

S. Kamvissis I On the Toda shock problem 245

The Dyson formula (A.1.30) is the starting point of our analysis (for a proof see the appendix). We further assume that the contribution to D ( n , t ) due to the part [ a - l , a + l ] of the spectrum is negligible. The reason of this is, of course, that the exponential appearing in (A.1.30) is bounded there, while it is exponentially large and hence dominating in [ - a - 1, a - 1]. Simplifying assumptions of this type were also made in the calculations of refs. [9] and [13].

We thus assume

R+---0 o n l a - l , a + l ] . (1.3.1)

The Dyson formula now becomes

b 2 = O(n + 1, t )O(n - 1, t) D(n , t) 2 ' (1.3.2a)

where

o o

D(n, t) = 1 + ~ I k , (1.3.3) k = l

with

Ik = f exp[tEQk(/~l, " '" , /*k; n, t)] d/z1 • • • d/z k ,

a~ (1.3.4)

and

a k = {(I .£1, ' ' ' , I.I.k)lJ,] ~. [ - a - 1, a - 1 ] ,

) = 1 , " . , k ; /'1<,u'2 < ' ' ' < / ~ k } . (1.3.5)

Here ,

kl(n Q A g I , " " , gk, n, t) = ~'~ t t- In z2(~'J)

j = l

1 1 ) + z(~,;) - z(~,i--- ] + 7 y(I , j )

1 In] z ( ~ i ) - z(~,) /=1 i = l

i#j

(1.3.6)

Note that y is a function depending on the reflection coefficient for the particular shock initial data. As it is of a higher order in 1/t it will not play any role in the asymptotic analysis and, in fact, it will be ignored from now on. Also

Z ( / / , ) = ~ - - a - - [ ( ~ - - a ) 2 - - 1] 1/2 (1.3.7)

where the square root is chosen to be negative when /x < a - 1.

We can write (1.3.2a) in terms of the variables x n. From (1.1.3) and (1.1.5) we can express x, in terms of the b , . Taking (1.1.8) into account, we eventually obtain

5c, = - 2 a t + In D(n - 1, t) O(n, t) (1.3.2b)

We have

Qk(IZl . . . . , /Xk, n , t ) = ( f , ~bk) + (Z~bk, d, ik) ,

where

n 1 f(~') = 7 In z2(~,) + z(~,)

Z( /z)

a - 1

1 I,nl I a--1 ( 1 . 3 . 8 )

and

'IT

¢,k(~,) = 7

Note that

k

6( / , - / , j ) . (1.3.9) j = l

q,~ --. 0 , (1.3.10)

and

a - 1

f ~b,(/z) d/~ = "7- a--1

for k E Z (1.3.11)

( the quantum condition).


Remarks:

(i) It is proved in ref. [13] that L is a negative definite operator.

(ii) It is clear that as t-~ ~, D(n, t) ~ 1 unless f is positive on some subset of the interval [ - a - 1, a - 1]. One can easily see that for

n v ~ V a + 1 ->N*-- t ln(x/-fi + x/-a--~) '

we have f < 0 on [ - a - l , a - 1 ] , while if n / t < N * , we have f > 0 on at least a subset of [ - a - 1, a - 1]. So for n / t > N*, b, ~ ½, i.e. the effect of the shock is not yet felt. On the other hand for n / t < N* the analysis of D(n, t) is non- trivial. From now on we restrict ourselves to the case N - n / t < N*. The above discussion shows in particular that the speed of the shock front is

x/-a V-d + 1 N* = . (1.3.12)

ln(x/~ + ~ )

Each I k is a Laplace integral which can be evaluated asymptotically by the Laplace method. In the spirit of refs. [9] and [13] we approximate each ~k by a non-negative qJ E L ' [ - a - 1 , a - 1], in the limit as t---~ ~.

Fk(n, t) = F(n, t) + [a(N)]k , (1.3.16)

where a is a continuous function of N = n/t and F is such that:

F(n - 1, t) - F(n, t) approaches a constant at large times. (1.3.16a)

Furthermore, it is easy to see that setting a ( N ) = 0 will only change our final answer up to a phase difference (cf. [13] section 1). From now on, we assume that indeed a ( N ) = O.

For a justification of theorem 1.3.1, see ref. [13]. Formulae (1.3.16) and (1.3.16a) are really assumptions, only partly justified in ref. [13]. In chapter 3 we will make a different assumption:

F k is independent of n . (1.3.16b)

Thus the problem of evaluating b, asymptotically is reduced to solving the following maximization problem:

maximize Q(qJ) for ~0 E A k . (1.3.17)

Writing

Theorem 1.3.1. For t large,

I k = exp max Q(O) + Fk , (1.3.13) ~b~A k

where j

Q(~0) = ( f , qJ) + (LO/, q0, (1.3.14)

and A~ is the set of all L ' [ - a - 1, a - 1] non- negative functions for which

1 1 q,= ?

with q~*, ~, ~ . . . . independent of t, we obtain

1 Q(q0 = ( f , ~b*) + (LqJ*, ~0") + t ( f + 2Lq~*, ~)

1 + ~ ( ( f + 2L~O*, ~) + (L~, ~))

1 ~- + 7 ( ( f + 2L~*, qJ) + (L~, ~)) + . . . I

a - - 1

= T '

a - 1

k ~ E . (1.3.15)

(Condition (1.3.15) is called the "quantum condition" in ref. [13].) The 'error' F k is a function of n, t such that, given 6 > 0, for N > 6,

The procedure is as follows:

(i) Leading order maximization. We maximize ( f , ~O*)+ (L~O*, ~b*) with respect to the condition qJ* >- 0. Note that the condition (1.3.14) is a higher order condition, so it does not affect the leading order maximization.


The variational condition for this problem is

f(A) + 2L~/,*(A) = 0 if $* > 0, (1.3.18a)

f(A) + 2L$*(A) -< 0 if $* = 0. (1.3.18b)

higher order terms are zero. We combine all the above to obtain

1 max Q(~b) = ½(f, $*) + 7~ (L~b, ~b). ddEA k l -

(1.3.22)

Calculations below (see section 2.1) show that the inequality in (1.3.18b) is strict:

f(A) + 2L$*(A) < 0 if $* = 0 . (1.3.18c)

(ii) First order maximization. We maximize ( f + 2L~b*, 6) with respect to 6, where ~b* is the solution of the previous problem, subject to

6(A) > 0 if qJ* = O,

So, writing 6k = 6 , we obtain

( t 2 1 (L6k, 6k) Ik = exp ~--~ (f, ~b*) + ~

(1.3.23)

For the definition and the properties of F k, see equations (1.3.16), (1.3.16a) and (1.3.16b) above.

a n d

a - 1

~ * + 7 d t z = k 7 ' a - -1

k~7 / (1.3.19)

Since ( f+2Ld/* ,6)<-O, we see by (1.3.18c) that its maximum is zero and is obtained if

6 = 0 when qJ* = 0. (1.3.20)

(iii) Second order maximization. We still need to find 6, if ¢* >0. By (1.3.20), 6->0 if ~,* = 0. The second order maximization now implies as above that ~ = 0 when ~b*= 0. In particular ( f + 2L¢*, d/) = 0. To complete the second order maximization we need to maximize (Lff, ~b) under the contraints

supp 6 C supp $* ,

and

2. T h e case a = 1

2.1. Solution of the variational problems

We begin with the solution of the leading order maximization problem. We only need to produce one solution ~b* of the variational condition (1.3.18). Then this wiU solve the leading order maximization problem uniquely, since L is negative definite and hence Q is strictly convex. Note that for a = 1, [ - a - 1, a - 1] = [ -2 , 0].

Theorem 2.1.1. The maximizer ~* of the functional Q(¢) = (f, ~b) + (L~, ~) such that ~* -> 0 and ~b*~ L1[-2 ,0] is given as follows. In the region N - n/t E [0, N*] we have

e(a) ~/,*(A) = - i R(A) A E [ -2 , 3"],

~*(a) = 0 a,~'[-2, 3'], (2.1.1)

a - 1

f ( 1 ) ~/,* + t 6 d/z = "-'f- a - - 1

k E 7 / . (1.3.21)

(iv) Higher older maximization. As before,

~* = 0 implies 6 = 0, and ( f + 2L~b*, ~) = 0. Maximizing (L~, ~) we find ~b = 0. Similarly all

where

R(A) = [(A + 2)(A - y)A(A - 2)11'2 , (2.1.2)

with R(A) chosen to be positive when A ~ 0% and

P(A) = (A - y)(A + N + 3,/2). (2.1.3)

248 S. Kamvissis I On the Toda shock problem

Here 3' is uniquely defined by

0

re(a) dA=O. Y

(2.1.4)

Proof: We make the ansatz that supp O *= [ -2 , 3'] for some y with - 2 < 3' < 0. The ansatz is justified a posteriori once we have produced a solution of the maximization problem, by un- iqueness. By differentiating (1.3.18a) with respect to A, we obtain

.y

1 / . [(i, 1) 2 1] 1/2~/*(/~)

N + a - 1 -

-2

when A ~ [ -2 , 3'].

= 0 ,

(2.1.5)

Also by our ansatz,

= o for a E [3", o ] .

Extending

is the Hilbert transform. We thus note that

~O(a) = - i P ( A ) [(A + 2)(A - 3')]1/z

= 0

Define 0 * ( a ) = ~(a) [ (X - 2 ) x ] " ~ "

Then

for a ~ [ - 2 , 3'],

for A ~ [ - 2 , 3 ' 1 .

- i P ( A ) qt*(A) = R(A) x e l - 2 , 3'1

= 0 A ~ [ - 2 , 3'], (2.1.10)

while (2.1.9) becomes (2.1.5). In other words, 0*(a) as given by (2.1.10) satisfies the differentiated variational condition for (2.3.17). Finally, it remains to determine 3'. One can see that if we substitute ~* given by (2.1.10) back into the undifferentiated variational condition we end up with

~b* = 0 for all A E R - [ -2 , 0], (2.1.6)

we see that ~b* is the solution of a Riemann- Hilbert problem defined by (2.1.5) and (2.1.6). This is easily solved as follows: let

0

f P(A)

Y

dA = 0 . (2.1.4)

We can solve (2.1.4) for N in terms of 3',

( P(*) F(A) = i A+ N - 1 - R(A)]" (2.1.7)

Then, clearly F ~ H p for 1 < p < 2. Hence by a classical theorem of Hardy (see ref. [8]), if we define

tp(a)Re(F(a)) , (2.1.8)

then

Hq,(a) = Im(F(a)) , (2.1.9)

where

1 f f(t.) Hf(A) = ~ ~ all* - o o

N = j'r° ( A - 7 ) 1 / 2 ( A + 3 ' / 2 ) / [ ( A + 2 ) A ( A - 2)11'2 dX

j,o ( x - 3'),,2/[(,~ + 2)x(,~- 2)l"2dX

(2.1.11)

One can easily compute that dN/d3" < 0 every- where. Also, one can see that as 3' ~ - 2 , N 1' N*, while as 3" 1'0, N ~ 0 . So N(3') is strictly decreasing and maps [ -2 ,0 ] onto [0, N*]. In particular, 3" is uniquely determined as a function of N by (2.1.11). This completes the proof of the theorem.

The discussion above shows how the support of $* varies with N: for N > N*, supp 0 *= 0 .

s. Kamvissis I On

In this case (see discussion in section 1.3) the perturbation of the b,'s is exponentially small and the effect of the shock is not yet felt. For N < N*, supp ~* = [ -2 , y(N)], while as N---~0, supp ~* tends to cover all of [ -2 , 0]. As we shall see later the motion of the lattice will be identified with an algebro-geometric solution corresponding to the continuous spectrum [ -2 , T(N)] U [0, 2]. For a fixed N this would be a time-periodic oscillatory motion. As N gets small, that is as the gap [y(N), 0] closes, we see decaying oscillations instead.

It is now easy to calculate (f , ~*). Combining (2.1.10) and (1.1.8) we get

( f , ~*) = A N 2 + B N + C , (2.1.12)

where

,y f ( X - - , y ) l / 2 I n z 2

A = - i [( ~'--~ ~ ' ~ = ~-~1 / 2 d* , - 2

(2.1.13)

7 f ( A - y ) ' / 2 ( z - 1/z)

B = - i [ ~ + 2)'~-A----2)~ dA - 2

T

- i f ( - 2

)[ __ , y ) 1 / 2 ln2(A + y/2) [ ( ~ ~ ' _ - ~]-i~ dA,

(2.1.14)

c = - i f ( a - 1/z)(A +y/2) [ ' ~ - ~ ~ i : 2)1 *'2 dA.

2 (2.1.15)

Each of the quantities A, B, C is a function of Y and hence of N, and can be explicitly expressed in terms of elliptic integrals and elliptic functions. As N--~ 0, we can say even more. From (2.1.11) we know that as 3,-->0, N-+0. An easy computation shows that

T = - 4N + ~ ' (N2) , (2.1.16)

the Toda shock problem 249

and expressions for A, B, C are

A = 2 In 2~r + l~r2N + ¢7(N2), (2.1.17)

n = -4~r + ]~rN - ]~r2N 2 +/7(N3), (2.1.18)

C = - ] ~ r N 2 + ~'tr. (2.1.19)

Hence, by (2.1.12), as N--~O,

1 (f , ~ . ) = ~ _ 4N + 2 In 2N2+ e(N4). ~r (2.1.20)

Next, we obtain the solutions ~k of the second order maximization problem. We need to maximize (L~k, ~k) under condition (1.3.15).

Theorem 2.1.2. The maximizer ~k of the functional (L~, ~b) such that

supp ~k _C supp ~* ,

~ , E L 1 ,

0

I( ' t " ~b* + 7 ~ , dy=-- - i - - 2

is given as follows:

- iE(k) ~k(A) = R(A)

=0

for A E supp ~*,

where R(A) is defined by (2.1.2) and E(k) is determined by (1.3.15):

iE(k) f dA "tr R(A)

- 2

where

.y

if = - ¢ , * ( a ) d a . ,IT

- 2

= of t - k , (2.1.22)

(2.1.23)

for AJ~'supp ~,*, (2.1.21)

2 5 0 S. Kamvissis / On the Toda shock problem

Proof. We only need to solve the unconstrained problem of maximizing the functional

.y

21( k), - 2

(2.1.24)

where 21 is a Lagrange multiplier. The variational condition satisfied by the maximizer $ is

N < N*, we have

F E H p for 1 - < p < 2 .

We are thus able to use Hardy's theorem once again. If

f(A) = Re F ( a ) ,

then

l (a) L~(A) = -- if A E [ - 2 , 3'1,

'1"¢

(b) 6(A) = 0 if A ~ [ - 2 , 3'],

-¢

1 f @(A) dA = k - trt (2.1.25) (0 ~ - 2

Differentiating (2.1.25a) with respect to A, we get

3'

J [(t~- 2)txl"~q,(t*) d/x = 0 1 - ~

- 2

for

X ~ [ - 2 , 3'1. (2.1.26)

Extending

qJ(A) = 0 for A ~ ' [ - 2 , 3'1,

we get

H ( [ ( A - 2)xll'2q,}(;t) = 0 for A E [ - 2 , 3'],

(2.1.27)

where again H denotes the Hilbert transform. Eq. (2.1.27) together with (2.1.25b) poses a Riemann-Hi lber t problem which again can be solved explicitly. Let

- i E F(A) (2.1.28)

[(A + 2)(A - 3 ' ) 1 1 / 2

Hf(A) = Im F(~t).

So let ~ = f ( A ) / [ ( A - 2 ) x l 1'2. Then,

- i E tk = R(A) for A ~ [ - 2 , 3'1, (2.1.29a)

~b = 0 for A J~'[-2, 3'1. (2.1.29b)

Also, Hf(a) = Im f(A) = 0 for h ~ [ - 2 , 3'1. Thus, ~k as defined by (2.1.29) satisfies the differentiated variational condition. The constant E is determined by (2.1.25c). We obtain

.y

iE(k) f dA R(x)

- 2

= o - t - k . (2.1.30)

This completes the proof of theorem 2.1.2.

It is now very easy to compute (L~b k, ~bk). We have

0

f dr L~k(A) = Ek R(Ix)

0

f = G R(~,) A

A E [ - 2 , 3'],

A ~ ' [ - 2 , 3'1. (2.1.31)

From (2.1.21), (2.1.22), (2.1.31) we finally get

(L~k, ~k) = i ~" ~r2(crt -- k) 2 , (2.1.32)

where E is a constant. As long as 3' > - 2 , i.e., where


fo da/n(a) = J'-*2 d A / R ( A ) " ( 2 . 1 . 3 3 )

Note that iz < 0, so that (L~k, ~k) < 0. We can now substitute these formulae back into (1.3.22) to get a formula for I k.

t 1 ) I k = e x p ~ (f , q,*)+ --~r (L~k' ~k)+Fk

~t [~ - 4 N + 21n2N 2 + ~'(N4)] + Fk} = e x p ( 1 z 3

× exp{i~tr(N)[tr(N)t - k]2}, (2.1.34)

where I-(N) is given by (2.1.33) and tr(N) is given by (2.1.23).

By the discussion following the statement of theorem 1.3.1, we see that, as long as we restrict ourselves in the region 6 < N < N*, then F k can be taken as independent of k (such an assumption will only alter the phase e 0 in the formulae appearing in the statement of theorem 2.2.1.). Hence, the factor exp(Fk) will drop off later when we consider the ratio D(n - 1, t)/D(n, t). Thus, we can ignore F k altogether.

Recall that D ( n , t ) = l + E k - ~ l k. We may write

oo

O(n, t) -- Y~ Ik , (2.1.35) - - o o

where I k is defined by (2.1.34). The resulting error will be exponentially small since the only terms contributing to (2.1.35) are those for which k ~ tr(N)t, in which case k is certainly positive, and ~?(t). From (2.1.34) and (2.1.35) we obtain, as N---~ 0,

D(n, t) = e x p ( 1 2 3 ~t [3 - 4N + 2 In 2 N 2 + ~(N4)]}

x ~, exp{i~rz(N)[~r(N)t-k]2}. (2.1.36)

D(n, t) = exp{ ½t2[A(N)N 2 + B(N)N + C(N)]}

x ~ exp{i~rr(N)[tr(N)t-k]2}, (2.1.37) k ~ - - c o

where A(N) , B(N) , C(N) are still defined by (2.1.13), (2.1.14), (2.1.15) which can be written in terms of elliptic integrals.

2.2. Identification with an algebro-geometric solution of genus I

Let 0(z[¢) = Ek=_~ exp(i~rck 2) exp(2~rikz) be the standard theta function. Then

oo

exp[i,rr¢(~rt- k) 2] = exp(i~r~-o-Zt 2) 0(ovtl~'). k = - - ¢ ¢

Eq. (2.1.37) now becomes

D(n, t) = exp{ ½t2[A(N)N 2 + B(N)N + C(N)]}

x exp(i~w(N)[cr(N)]Zt 2} O(cr(N)z(N)tl~'(N)).

(2.2.1)

When N is small,

D(n, t) = exp{ 1 2 z ~t [~ - 4 N + 2 In 2N 2 + ¢7(N4)]}

× exp{i~r~'(N)[o,(N)12t ~} O(cr(N)'r(N)t[z(N)).

(2.2.2)

On the other hand, Jacobi 's transformation gives

Hence, for N small,

D(n, t) = exp{ ½t2[~ - 4N + 2.1n 2N 2 + ~(N')]}

x ~ ¢ ~ N ) O ( t r ( N ) t l - ~ ) " (2.2.3)

In general, for N < N* not necessarily small, we of course have similar expressions

We obtain an additional expression for tr(N) by use of the Riemann bilinear relations.


Proposition 2.2.1.

o

-r 2

(2.2.4)

Proof. Consider the Riemann surface corresponding to the function V~-(A) = "ff(A + 2)(h - 3')(A - 2) which we visualise as two sheets glued together along the bands [ -2 , 3'] and [0, 2] in the usual way.

Let a be the cycle encircling [ -2 , y] and lying on the top sheet, while/3 encircles [3', 0] and lies partly on both sheets, as shown in fig. 1.

Let to = ~*(A)dh. Then

i f o- = 2"4"1T t o ~ ~t

while (2.1.4) becomes

f t o - - - ~ 0 .

upper and lower sheets respectively. The lower limits in the integrals of the RHS are irrelevant. We then have,

0 oo

2o- R(A) - N + 2 , "y oo'

which is equivalent to (2.2.4), since obviously

; f dA = 2 dA R(X) R(A) •

oo, 2

This completes the proof of proposition 2.2.1.

When N is small, the asymptotic evaluation of (2.2.4) shows that

2 o- = -- + O(N), (2.2.5)

and similarly, from (2.1.33),

Note that dh/R(A) is the differential of the first kind, while to is a differential of the third kind. The Riemann bilinear relations (e.g. [1]) then give

f d;t _R_cz f w f R(A)

= 2~i(iN) R--~ + 2~-i(-iN)

x ~ + 2 ~ i - i . ( - 1 ) + 2 ~ i ' i . ( - 1 ) .

Here do and oo' are the points at infinity in the

Fig. 1.

,IT r = i ~ + ~7(N). (2.2.6)

From (2.2.4) we obtain

1 I; dX/R(a) o-(N)t = io dA/R(A) t Iv ° dA/R(A) n .

(2.2.7)

Each of the integrals on the RHS depends on 3' and hence on N. Consider

( 1 I7 OX/R(X) I @(N, n, t) =- 0 j-o dA/R(A) t - j-o dA/R(A) n

1 r(N)) " (2.2.8)

Detailed computations show that

( , ) O N - 7 , n - l , t = O ( N , n - l , t ) + ~ ? ( N z) (2.2.9)


uniformly in n, t. Thus, using (1.3.2b), (2.2.3), (2.2.7) and (2.2.8), we easily compute for N small:

Theorem 2.2.1. Let 8 be any small positive num- ber. Then, in the region 8 < N, when N is small, we have as t---~oo,

O(N, n - 1, t) ) x, = (1 - 2n)ln 2 + In O(N, n, t) +/7(N2) '

(2.2.10)

1 n x,(t) = - 2 In 2n + ~ -~ [1 - cos(4t - "n'n + e0) ]

+ ~ (N2) , (2.2.12)

while using (1.1.3) and (2.2.10) we obtain

b 2= O ( N , n + l , t ) O ( N , n - l , t ) [e(N, n, 0] 2 + ~ ( N 2 ) "

(2.2.11)

We can now identify the solution of our problem with a modulated algebro-geometric solution.

1 n b,(t) = 1 + ~ t cos(4t - qcn + e0) + e ( N 2 ) ,

(2.2.13)

2 n an(t) = 3 t s i n ( 4 t - ~rn + e0) + •(N2),

(2.2.14)

where e o is some constant.

Proposition 2.2.2. Let

[ O(N, n - l , t)'~ X ( N ' n ' t ) = ( 1 - 2 n ) l n 2 + l n ~ O-(-~,n,-t) / "

Then X(N, n, t) considered as a function of n, t depending on the parameter N, solves the Toda equations (1.1.1) (together with Y(N, n, t ) = ~'(N, n, t)). Similarly, let

O(N, n + 1, t)O(N, n - 1, t) B(N, n, t) = tO(N, n, t ) ] 2

and A(N, n, t) = - ½,~(N, n, t). Then A and B solve the Toda equations (1.1.4a,b). In fact they form a one-gap algebro-geometric solution com- ing from a Riemann surface of genus 1.

Proof. The proof follows directly from the theory of algebro-geometric solutions of the Toda lattice. By eqs. (2.2.1) and (1.3.2) one gets formulae similar to (2.2.8), (2.2.10) and (2.2.11), where the theta functions involved have period ~- (instead of -1/~-). Then, one only needs to observe that the coefficients of n, t defined by the integrals in, say, (2.2.8) are exactly identical to the ones on the Da te -Tanaka formula (see ref. [3]).

Proof: The proof follows directly by expressing ratios of theta functions in terms of elliptic functions. We omit the details. Note that the phase e 0 appears if we do not discard the F k in formula (2.1.34) (see discussion there).

Of course, using arguments similar to the above we can extract a,(t) and b,(t) for any N < N*, N > 8, as t---> ~. We then get (for a,(t) say)

a , ( t ) - - X ( t ) s n [ t o ( t ) t + e ( t ) ] , (2.2.15)

where X, to, e, and the modulus of sn are given by expressions involving elliptic integrals.

Numerical computations suggest that for any fixed n, theorem 2.2.1 in fact holds for any large t, provided n is not too small. Eqs. (2.2.12), (2.2.13) and (2.2.14) then express the behavior of a fixed (the nth) particle for large times. The "leading order" behavior of x , , b , , a , is of course:

x , ~ - 2 1 n n , b , ~ l ,

as t -* oo, for all n, since the quantum condition is only effective to higher orders.


3. The case 0 < a < l N---~ N , -= 1 - a . (3.5)

In the subcritical case where 0 < a < 1, we have

(i) spec(L(t)) = [ - a - 1, a + 1] where in fact the band [a - 1, - a + 1] has multiplicity 2 with the rest of the spectrum having multiplicity 1. We also assume, as before

(ii) R+ - 0 on [a - 1, a + 1] (see (1.3.1)). We still have to solve the maximization prob-

lem described in section 1.3 and thus a corresponding variational problem.

Theorem 2.1.1. still holds with a few changes as follows:

qJ* ~ L l [ - a -- 1, a - - 1], (3.1)

and

R(A) = [(a + a + 1 ) ( A - y ) ( a - a + 1)

x ( A - a - 1)] l/2 . (3.2)

Here y is uniquely defined by

This means that the discussion in section 2 is valid only for 1 - a < N < N*. We can thus obtain an asymptotic expression for the solution identical to (2.2.15) with appropriate expressions for X, to, e, in that region. For N < 1 - a the above analysis fails. The quantum condition is no more relevant and we cannot capture the oscillations that (as we know from numerical experi- ments) still occur. However , we are still able to obtain the leading order behavior of the lattice.

Theorem 3.1. a,--->O and b,---~ ½(a + 1) as t---> oo.

Proof . T h e leading order maximizer (see p. 10) qJ* of Q(q0 = ( f , qJ)+ (Lq,, q0 is now given by

A+ ½(a- I)+N ~O*(a) = - i

[A = - (a + 1)21 l/2 '

A E [ - a - 1, a - 1]. (3.6)

(The proof simply copies the proof of theorem (2.1.1).) We also have

a - 1

r e(a) dA = O. (3.3)

The proof goes through as in section 2.1. The difference is now the following: while in the case where a = 1, we see that as 3,--+0 (i.e. as the gap [% 0] closes), we have N--~ 0, in the case where a < 1, the gap closes at a particular value for N, say N . > 0. We have

N = -

a - - 1

f (A - y)l/2(A + ~,/2) 1)]w = dA [(A + a + 1) (A- a + 1 ) ( a - a -

y

5--x

f (X_ y),,2 [(A + a + 1)(A - a + 1)(X - a - 1)p '= da "r

(3.4)

and as y - * O ,

Q(qJ*) = ( f , q,*) + (Lq,*, qJ*) = - ( j r , qJ*)

(by 1.3.18a) and thus Q(~O*) can be evaluated by computing integrals as in section 2.1. We obtain

! ( f , I11") = ¢1 + c2N + 2 ln(a + 1)N 2 + / ? ( N 3 ) , q'¢

(3.7)

imitating the proof of (2.1.20). Here c I and c 2 are constants whose exact value is irrelevant.

The second order maximization problem (see section 1.3) now leads to

' Ld/k(A) = 1 for A E [ - a - 1, a - 11,

a - - 1

t - - a - - 1

It turns out that (3.8) admits no L 1-solution. If it


did we would be able to obtain it by solving a variational condition (as in theorem (2.1.2)). However the solution we get in such a way is not L I.

On the other hand we have

4. The generalized Toda shock problem

We consider the Toda equations

2 a . = 2 ( b . _ t - bE),

sup(L~O, q~) = O,

where the supremum is taken over functions with support contained in the support of q,* and such that condition (1.3.21) holds. To see this consider the sequence

-n q,~(x) =, E.'

X/(A + a + 1)(a - y.)(A - a + 1)(A - a - 1) '

where % is some sequence converging to a - 1, with y, < a - 1, and E~ are constants chosen such that (1.3.21) is satisfied. We observe that 67, ~ Ak and that (L~b~,, ~O~,)--->0 (see proof of theorem 2.1.2). (Note that the ~ do not have a limit in LX.)

We then obtain

/2 D(n, t) ~ exp ~- [c 1 + c2N + ln(a + 1)2N 2

+ eCN')] + E , k = l

(3.9)

where the functions F k are defined in the statement of theorem 1.3.1. By assumption (1.3.16b) F k is independent of n. Thus, combining (1.3.2a) and (3.9) we obtain:

(2bn) 2 = (a + 1) 2 + e ( N ) ,

and since b, is always positive,

= ~ + ~ as t---> o0.

It follows immediately that the velocities

an'--'>O as t---> o0.

This completes the proof of the theorem.

b n = b,,(a n - a n + l ) ,

with initial conditions

a , (0) = a} bn(O) sgn(n)~ for large Inl.

In this section we consider all possible values of n, since the symmetry relations (1.1.5) no longer hold. The following analysis goes through for all n .

~ e c ~ e a > l

The continuous spectrum of L still consists of the bands [ - a - 1, - a + 1] and [a - 1, a + 1]. In general, however there will be several discrete bound states (a compact perturbation will not alter the continuous spectrum of an operator but may alter the discrete spectrum). Suppose then, that the discrete spectrum A consists of p eigenvalues

A = {A1, A 2 . . . . , At, } .

In this case, the Dyson formula is (see ref. [131)

(2b,)2 = D(n + 1, t )D(n - 1, t) [O(n, t)] 2

where

b 1 D ( n ) = l + • -~.

k=12

× f " " f d V ( Z l ) d V ( z 2 ) " . d V ( Z k )

× ( z , . . . :-+:

x II~-~<J'~k (Zl - zj) 2

I ' l l~ i , j< k (1 - zizj) '

2 5 6 S. Kamvissis / On the Toda shock problem

and

dz I d r ( z ) = R + ( z ) exPt(z - z- ') t 1 2--~--ff It,t=1-,

I z -1 - ~ l i t + (z)l ~ , [ z - (z-~ --~'-- z _ ( z ) -

dz x exp[(z - z - 1 ) t ~ [~1, ~zl

P

+ ~ a i exp[ (z+ , i - z-+ l , i) t] i=l

x ~(z - z+,,).

H e r e a~ are the no rming constants for the eigen- va lues A~'s and

z+ = A - a - ~ / ( A - a) 2 - 1 ,

z _ = A + a - ~ / ( A + a ) 2 1 ,

z+,i = z+( X3 ,

z 1 = - ( 2 a - l + ~ / ( 2 a - 1 ) 2 - 1 ) -1 ,

P

D ( n , t) = 1 + ~ , t~ i exp[ tZQl(A,) l i = 1

+ ~ ~%exp[t2Q~(x. xj)]+ l~i,j~--p

" " + = -~. f " " e x p { t E [ Q k ( ~ l " '" ~k)]}

X d/~l • • • d/.L k + ot i k~ " i = 1 k = l

P ~ 1

x d~l . . , d ~ + ~ ' ai~i ~1 ~ I , ] =

× dlZl • • • dl~, + • • •

where

Q k ( l X l " ' " tXk)

= ~ l ( n 1 ) ./=1 7 7 ln[z2(/xJ)] + z(/JT) Z(/Xj)

1 in I z ( I z , ) - z ( l ~ j ) )

i # i

(4.1)

z2 = - (2a + 1 + ~ / (2a + 1) 2 - 1) -1 .

O n c e m o r e we assume R+ - 0; we also no te tha t if A > a + l , z + ( A ) - [ z + ( A ) ] -1 is negat ive .

H e n c e it is only e igenvalues Ai with a~ < a - 1 tha t con t r ibu te to the third t e r m of d r ( z , t). F r o m now on we consider only

A i < a - 1 .

I t is easy to see tha t

I2Qk+I( I£1 " ' " lZk, A)

, , ) + H(),) + e ( ~ ) , t 2 Q k ( ~ l . . .

where

2 H ( A ) = f ( h ) + 2 L ¢ * ( A ) + t L~bk(A) '

We will also assume, for the m o m e n t , tha t there a re no e igenvalues be low - 1 - a , so tha t

A i > l - a .

T h e D y s o n fo rmula can be wri t ten as follows, a f te r s o m e trivial computa t ions :

wi th f , ~* , @k as def ined in sect ion 1.3. Also,

t2Qk+2(l£1 " '" ~k , Ai' Aj)

= t 2 Q k ( l ~ l " ' ' l ~ k ) + H(AI) + H ( A j ) + ~7(!2) ,

and so on, for t2Qk+p and any p -> 1.


We evaluate the integrals in the Dyson formula above, by solving maximization problems as in section 2.1. So, for example, the first integral of (4.1) can be written as (compare with (1.3.23))

- a + l

- i f A2+C ~/'= ~r Ro(A ) d a ,

- a - - 1

Ro(A ) = [(A 2 - (a + 1)2][A 2 - (a - 1)2)] u2 .

t 1 ) I , = e x p 2-~ ( f ' O*) + - (L~bk' ok) + '

We can now see, after some computations (similar to those of sections 4 and 5 of ref. [13]), that as t---> % D(n , t) can be written as follows:

where in fact F k can be ignored (see section 2.1). Furthermore it is easy to compute (see, for example, section 4 of ref. [13]):

1 (LqJk, qJk) i~r~'( n )2 ~r ~ + t o t k ,

( f , qJ*) = ln(4a)N 2 - 4 a N ,

a - 1

- ~ i ( t r t - k) f dX' L~bk(A) = f _ , ,+ , d A / R o ( A ) Ro(A, ) ,

J--a--1 A

and

p + l

D(n , t) = ~ D , ( n , t) , (4.3) i = 1

where

D , ( n , t) = O(½n - totlr) P

+ E eA" -s "O(½n - t o t + e i lz ) , i = 1

D2(n, t) = ~ exp[fi +f~) . - (gi + gj)t] i = 1 j = l i<j

X O ( ~ n - to t+ e i + ej l , r ) ,

(4.4)

(4.5)

a - I

H(A) = 2 d/z /z + N/z + c + 2¢, (a1 A

+

and so on. Furthermore, one sees that

A i

f d, e i = 2 --,--,Ro('~ . - a + l

(4.6)

where c is defined by

a - 1

f A 2 + c = 0 . a + l

Here

dX/Ro(X) 1" = a~-if-"+l d A / R o ( A ) ,

o ' = - ½ N + to,

(4.2)

Remark . Although one can find expressions for f~ and gi in terms of elliptic integrals (easily for small p and in principle, for every p) it is hard to find a general formula holding for all p. We will show later how to circumvent this obstacle.

The above formulae give an expression for the asymptotic solution of the (general) Toda shock problem. Theorem 4.2 shows that we obtain the same expression from a Da te -Tanaka algebro- geometric solution of genus p + 1 by shrinking all but the two outer bands to a point.

We first present the main result of ref. [13] in a convenient way.

258 s. Kamvissis / On the Toda shock problem

Theorem 4.1. The solution of the special Toda shock, i.e. the Toda equations under

a ,(0) = sgn(n)a, n # 0 ,

a 0 ( 0 ) = 0 , b , ( 0 ) = ½ ,

is asymptotically described as follows. (i) For n/ t > N*, a, and b , - ½ are exponen-

tially small. (ii) For N* > n/ t > Nmi n (for some particular

value of n / t - N = Nmi n defined in [13]), a.(t) and b, ( t ) perform single-phase modulated oscillations•

(iii) When n/ t < Nmin, a,( t ) and b~(t) perform "binary" oscillations. In fact they form a degenerate 2-gap algebro-geometric solution where the middle band is shrunk to a point, of band structure [ - a - 1 , - a + l ] U { A o } U [ a - l , a + l ]. []

Our main result is the following.

Theorem 4.2. Suppose A t . . . . . Ap are the eigenvalues of L and [ A i l < a - 1 . Then, when n / t < N.,i,, a , ( t ) and b, ( t ) form a degenerate (p + 1)-gap algebro-geometric solution where all but the two outer bands are shrunk to a point, of band structure [ - a - 1, - a + 1] O {A~} O . ' . O {kp} O [a - 1, a + 1]. On the other hand, when n/ t > N*, a , ( t ) and b , ( t ) - 1 are still exponentially small.

Proof o f Theorem 4.1. (i) and (ii) and proved in [13]. For (ii), we have to identify the Darboux-

transformed period-two solution in [13] with an appropriate degenerate algebro-geometric solution.

Consider a genus 2 Riemann surface corresponding to the band structure

[ - a - 1, - a + 1 ] U [ - e , e ] U [ a - 1, a + 1],

where e > 0 is small, in the sense described in section 2. Let

R.(A) = (A 2 - e2)(A 2 - ( a - 1) 2 )

x (X 2 - (a + 1)2),

and

W 2 ~ - -

Clo dk cl~k dA + - -

c20 dk c21A dA

be the corresponding Abelian differentials of the first-kind normalized by

f wj = ~q, i, j = 1 ,2 , a i

where a 1 and a 2 are the cycles shown in fig. 2.

Also, let

eo

c j ~ J wj , 00'

- o - 1

1~2

Fig. 2.


where oo' is the infinity point on the lower sheet of the Riemann surface. We easily compute, as e--~ 0, up to an error ~7(e),

i(a 2 - 1) i a + 1 Cl0 = g ' 27 ' cH = 4

i a + l c 2 0 = 0 , c21= 2 K ' '

where

(4.7)

K'= K(2' \ a + l / '

K being the elliptic integral of the first kind. Also

i In(a) i K K cl = 2~r 2 K ' ' c 2 = - i ~-7 , (4.8)

where

a - 1

Let

Tij = f Wj , aj

with fli as shown in fig. 2. We compute, as e---* 0 ,

(4.9a) i l n [ 2 / a 2 - 1 ~ + i K Tll = ~ \ ~ \ - - - - ' ~ - - - ] , ] ~ "1- e ( e ) ,

. K z12 = 1 ~7 = ~'21 , (4.9b)

K r22 = 2i ~-7 • (4.9c)

The Da te -Tanaka 2-gap algebro-geometric solution corresponding to the above structure is given by (see ref. [12], p. 129)

(2b.)2 = O(n + 1) a(n - 1) [a(n)]2 , (4.10)

where O(n) is the (2-dimensional) theta function

O(n) = O(nc -- c ' t + 8 ) , (4.11)

and

c = c2 , \C2o/, 82

where $ is some phase constant (expressed explicitly in ref. [12]). Furthermore, we have that, as e "--> 0 ,

82 = (7(1), 81 = 1~- u + ~7(1).

Here, the period matrix for (4.11) is (zq) defined by (4.9).

Let

z i = nc i - Cil t + 3/, i = 1, 2 . (4.12)

Then

O(n) = ~ exp(E~-im2z2 + ~-im2r22) F ( m 2 ) , m2

where

F(m2) = ~ ' exp(2~rimlz I + 2~rimlm2~'12 ) nl l

X exp(~rimlz,rll)-

As e--*0, since ~ ' u = - i / ~ r l n ( e ) + ~ ( 1 ) , the dominating contribution to F(m2) comes when m l = 0 0 r m t = l . We thus get

F(m2) ~- 1 + exp(2~ir12m 2 + 2,rriz 1 + ½,triTll/2 ) .

Hence,

o(n) --- O(z&22)

+ exp(2~iz I + ~i7"u/2 ) O(z 2 + ~'121~'22).

Combining (4.7), (4.8), (4.9b), (4.9c), (4.12) we get, choosing the phases $ appropriately,

O(n) ~ 0(Z2]'r22) + a nO(z2 + ½,r221z22) ,


which together with (4.10) gives the Dabroux- transformed 1-gap solution as expressed in ref. [13]. This completes the proof of theorem 4.1.

R e m a r k : T h e degenerate algebro-geometric solution described above is periodic in time, and periodic in space for large [nl. For small n, we see the effect of a "breather": the amplitude of the oscillation is somewhat smaller.

P r o o f o f theorem 4.2. T h e proof follows that of theorem 4.1. We now consider a Riemann surface of genus p + 1 corresponding to the band structure:

[ - a - 1, - a + 1 ] U [ A 1 - e, A 1 + e ] . . .

lap - e, Ap + e I U I a - 1, a + I l ,

where e > 0 is small. We have

R,(A) = [A 2 - (a + 1)21[A 2 - (a - 1)21

P

x I-[ - - - +

We define

normalized by

f w~ = a o , aj

i, j = 1 . . . . . p + 1. (4.13)

The homology cycles a,. and /3~ are as shown in fig. 3.

The periods 1" u are defined by

/- 7 o • = J w j , i, j = 1 , . . . , p + 1, (4.14)

and

Cj = f Wj , oo'

where 0% o0' are the points at infinity on the upper and lower sheets of the Riemann surface, respectively.

The Date -Tanaka algebro-geometric solution of genus p + 1 is given by ([12], p. 129): equation (4.10), where each 0 is a (p + 1)-dimensional theta function. Setting

i cjiA dA W] =

/ = 0

Z i = nc i -- Cipt-F 8 i , i = 1 , " . , p + 1 ,

(4.15)

-a-.-1

I3p+1

i , , ~ t U'P+I

\ !

\ / \ /

\ / J

Fig. 3.


we have

p + l p + l O(n) = O((z,),= 1 I(%),,j=l)

=- E ml ," "mp+ 1

where (m / ltn ~

\rap+l/

exp(2~im • z + ~rim~'m'),

l! 1) Z m=

g +1

/, ~ p + l and ~r = O','jn,j=l. Then

(4.16)

o(n) = o(z,,+11~',,+1 , ,+ , )

P

+ ~ , exp[2~i(nci - cipt)] i=1

X O(Zp+ 1 + q' i ,p+l[q'p+l,p+l)

P

+ ~ exp{2~i[n(ci + c i) - (Cip + Cjp)t]} i=l j=l i<j

x O(zp÷l + ~i,p÷1 + * , ,p+ l l%+1 ,p÷ l ) + "'"

(4.17)

Remark: Again it is possible in principle to calculate c~ and Cip in terms of elliptic integrals, although such formulae are very complicated for large p.

(2b2) = O(n + 1) O(n - 1) (4•10) [ e (n) ] 2

We now have to examine (4.10), (4.15), (4.16), as e--+O. We can easily see from (4.13), (4.14), that, as e--+O,

i ~'jj = - - - l n e + ~ ( 1 ) , ~ r j < p + l

We now compare (4.17) with the similar expression for D(n, t) in the solution of the general Toda shock problem: (4.3)-(4.6) . It is easy to check that, apart from a possible discrepancy between f//(as defined in (4.5)) and 2~ric i on one hand, and gi (as defined in (4.5)) and 2"nic~p on the other, the two expressions are identical.

To complete the proof of theorem 4.2 it remains to show that

and fi -- 27rici , (4.18)

~'ij = 0(1) for i # j ,

while

• K ( ( a - 1)/(a + 1)) T p + l , p + 1 = I r (2x / -d / (a + 1))

where K is the normal complete elliptic integral of the first kind. Also (by p. 126 of [12]), as E " ~ 0 ,

8j = ½~-q + ~7(1).

As e -+ 0, (4.16) splits into a sum of one-dimensional theta functions. Choosing the phases 6j appropriately, we obtain

gi = 2*ricip • (4.19)

We postpone this until the end of this section• But first, we describe the qualitative features of the solution of the general Toda shock problem (or equivalently, the corresponding degenerate algebro-geometric solution). For simplicity consider first the case p = 1, with A 1 # 0 . When p = 1, it follows from (4.4) that

D(n, t) = O( ½n - tot)

+ exp(f ln - glt) O(½n - tot+ 6 1 ) ,

D(n + 1, t) = O ( l n - to t+ ½)

+ expt / l (n + 1) - glt] O(½n - tot + e I + ½ )•


(Recall that x,+ 1 = - 2 a t + In D(n, t ) / D ( n + 1, t).)

One can see that when )t 1 = O, we have gl = O, while when )q < 0 , we have g~ > 0 , and when )q > 0 we have g~ < O. Also, f~ > O.

Suppose first ~1 = O. Then as n---~ ~,

D(n, t) O( ½ n - to t + e x) e x p ( - f l ) •

D ( n + l , t ) O ( ½ ( n + l ) - o J t + e l ) (4.20)

We thus obtain (see [3]) a solution of the 1-gap solution with band s t ructure [ - a - 1, - a + 1] t_J [a - 1, a + 1] (which is periodic in space and time). As n---~-0%

D(n, t) O(½n - tot) (4.21)

D ( n + l , t ) O ( ½ n - t o t + ½ )

We obtain the same solution with a phase shift. Around n = 0, we notice that the amplitude of the oscillations is smaller than when Inl >> o. We thus have a breather.

Remark. The solution described above is of course the solut ion of the special Toda shock problem.

Next suppose h 1 < 0 . Then, as n/t--->~, we again obtain (4.20), while as n/t--->-0% we obtain (4.21). Once more we observe the same phase shift. In fact we can see that the solution is

periodic on the regions n/ t >> gl/fl and n/ t ~ gill , while as n / t ~ g l / f l we see a travelling soliton (with s p e e d = gl/ f l ) . The situation is analogous when h I > 0. Similarly, in the general case with p eigenvalues, we see that the region N < N* of the n - t space is split in different regions, separated by solitons. For each eigenvalue, we see a travelling soliton (provided h i ~ 0, otherwise we see a breather) , while as we cross the boundaries between regions we observe phase shifts (which can be calculated in terms of e~, as defined in (4.6)). Also the speed of each travelling soliton can be expressed in terms of f~, gi.

We finally give a justification of (4.18) and (4.19). We note first that the coefficients c i and Cip depend only on the continuous spectrum of the solution, and hence on a alone.

One can easily check that the asymptotic expression for the solution of the Toda Shock problem as extracted, solves the Toda equations up to an error ~7(1/t 2) only for specific values of f~ and g~. The fact that the algebro-geometric solution (4.10) solves the Toda equations exactly shows that, in fact, f~ and gi have to be given by (4.18) and (4.19). This completes the proof of theorem 4.2.

So far we have been assuming that all eigenvalues lie in the gap [ - a + 1, a - 1]. The effect of eigenvalues h with h < - a - 1, is seen in the region N > N*, before the shock front is felt. In that region, we can see that for each eigenvalue we have a "s tandard" travelling soliton (that is, in vacuous background). The solution can be identified with a degenerate algebro-geometric solution where every band is shrunk to a point, one band for each eigenvalue. The relevant computations can be found in, say, ref. [10].

The cases a = 1 and a < 1

In these cases there is no gap between - a + 1 and a - 1 so the situation is trivial. Any eigenvalues correspond to "s tandard" travelling sol- irons.

Acknowledgements

The author wishes to thank Percy Deift, Stephanos Venakides, Peter Lax and Henry McKean for helpful suggestions and comments. This work was partially supported by a Fellow- ship from the Onassis Foundation of Greece.

Appendix

A. 1. Inverse scattering and the Dyson formula

We present here the basic facts concerning the


scattering data, their evolution, and the solution of the Toda shock problem in terms of the scattering data. We follow closely the corresponding t reatment for the supercritical case in [13], with a few alterations at crucial points. We assume 0 < a - 1. We consider symmetric tri- diagonal matrices

( ) • a_ x b_l M = b_: a o b 0 ,

b0 al bl • . " .

(A.I.I)

where

an---~a as n----~+oo, (A.1.2)

a , - - - * - a as n----)-o0, (A.1.3)

b.--+ ½ as I n l ~ (A.1.4)

rapidly (faster than any polynomial). The continuous spectrum of M consists of the band [ - a - l , a + l ] . The part [ a - 1 , - a + l ] has double multiplicity, while t he rest is of single multiplicity• If a = 1, the band [ - 2 , 2] has multiplicity one. In the special ease defined in section 1.1, there is no discrete spectrum. Let

z+ = A - a-V(A-- a) 2- l, (A.I.5)

and

z_ = A + a - ~/(A + a) 2 --I, (A.1.6)

where the square roots are positive as A---> +w. Variable z+ maps C - [a - 1, a + 1] injectively onto 0 < z + l z + l < l , while z_ maps C - [ - a - 1, - a + 1] injectively onto 0 < ]z_] < 1.

The Jost solutions f,+(A) and f~ (A) are defined by

Mf~ = Aft, (A.1.7)

and the asymptotics

+ n f n ( Z + ) ~ Z+ as n---> +co, (A.1.8)

f - , ( z _ ) - z - " as n--- , -oo. (A.1.9)

The f ~ ( z + ) are analytic in ( 0 < l z + l < l ) and have continuous derivatives of all orders up to Iz+l = 1. Similarly with f 2 ( z _ ) .

On Iz+l = 1, fX(z+) and f+(z+) a r e two in- d e p e n d e n t solutions of (A.1.7); T+ and R+ are defined by

T+ f-, = R+ f+, + f+, , Iz+l = 1. (A.1.10)

Similarly

T + , • _f = = R_f: + ~ Iz_ l = 1 (A.I.II)

In the special Toda shock case we can easily see that

2 a + z + - z = _ , (A. l .12) R+ 2a + z+' - z_

and

I r _ l 2 = (1 - ( , + a) 2)

- 4aA ' (A.! .13)

when t = 0.

Also, the evolution of these scattering data is given by

R+(t) = R+(0) exp[(z+ - z+l)tl, (A.l .14)

I T_(t)l 2 = I T_(0)I 2 exp[(z+ - z~a)t], (A.l .15)

whenever z+ is real. Next, we write

f.(z.) = (2b~)- ' u . ( z + ) z 2 . (A.1.16)

Theorem A.1.1. u . ( z+) is the solution of the "Marchenko- type" equation


co

Un,k'+ Wk+2n + E Wk+j+2nUn,j = 0 , j = l

k = 1 , 2 , . . . , (A.1.17)

where u~, k is defined by

d o

"n(Z+ ) = E " . , i z ~ , 1=0

(A.1.18)

Wk = Pk + Tk ' (A.1.19)

2 ~

dO Pk = eik°R + (ei°) ~ ,

o

(A.1.20)

z o

-1

I z ; ' - z + l z ~z: dz+ [z-l---z"~_i IT-( +)1 + 2~lz+i '

(A.1.21)

Zo= Z + ( - a - 1 )

= - 2 a - 1 - ~ / ( 2 a + 1) 2 - 1 < 0 .

Also,

0o ao

I-I(~b + X + k) -1 = 1 + w2. + W / + 2 n U n , / • n j=l

analytic in {Iz+l < 1 ) - {-1 , Zo}, we have

dz T+f-~ Z+ +k 2~riz+ B+

f dz T+f-~ n+k = 2"triz+ n+ Z+ .

Iz+l=~ (A.1.22)

Near z+ = 0, we have

f-~(z+ ) = B~zT"[1 4- e(Iz+l)],

and

-1 2 T+ z+ - z+ 1 - z+

= 2 w " = I t " [ 1 + e(lz+l)].

He re,

n - 1

B n = N ( ½ b k ) -1 ,

B += rI (½b)- ' n k k=n

o0

B = I-I ( l b k ) - i k= -0o

(A.1.23)

(A.1.24)

(A.1.25)

and W is the Wronskian

Proof: Consider the following shown in fig. 4.

Applying Cauchy's theorem,

contour C as

since T+ is

- t " -l+e z:) 0

W = + b~(f~+,f - f+~f2+l) . (A.1.26)

The right hand side of (A.1.22) is

R H S = f

Iz+l=~

dz÷ B : z ; o ° + ,

--..-X-7---Z+ 2"triz+ \ B / B

2 ~

f da z+k ( 1 - z 2) o 2~ (B:)~ 2~ f dt~ e ika o 2~r (B+) 2 ego + ¢7(lel))

0 k > 0 . -- 1/(b~+) 2 k = 0 .

Fig. 4. On the other hand the left hand side of (A.1.22)


consists of two terms. First

f d a exp[i(n + k)a] T + f-~(ot)

B +

f da R+(a) + + fn(Ot) f+(a) = ~ "-:w

-~r Bll

Writing

p m ( O / ) e ,

and using

/:(~) = B + ina U e ij°t ,. e - n , j - +1 ,

we have

exp[i(n + k ) a ] .

(A.1.27)

.-1

and

T_ = ~

C

_ - 1 4 - ~

>

Fig. 5.

- 1 Z _ - - Z

2W

,0

"Y,u

f 0

da + e-ima ina eiia ~n Pm e u~, i + 1

+ e -"~ u., j e -q= + 1 exp[ia(n + k)] " 1

E + = Unk + P 2 n + k + j U n / "]- P 2 n + k " j = l

So

- T_f~ - T_f: = - 'l'_R_f~ - T_f~

= - I"_(R_f: '+f--~)

T 2 +. =-?-T-fn++-I -Ifn

Thus

The other term in the left hand side is

Zo

f dz+ T+f- , - T + f : n+k - 2~iz+ B + z+ ,

- I

(A.1.28)

where z+ = z+(A+); A + is meant to be on the upper half-plane, hence z+ is in the lower half- plane (see fig. 5). We have

T+ z : l - z+ --1 T .

z _ - z _

Also

Z+

× (T+/-~ - T+f:) = -T_f: - T_f:

T + f : T + f \ Z :1 Z _

Integral (A.1.28) then becomes

Zo - T 2 + - I-I . k+n __ f dz (z_~+_-11 ~-~)(A+) z+

f[o dz T 2 k+2~Iz+1-z= I = ~ _ z + z : * - z _

~- "l ']+k+2nUnj + 1 1

where 7 k is defined by (A.1.21). Substituting


back into (A.1.22) we get

ao

+ ~ + ilnk "~ Wk+2n "~ Wk+j+2nUn] = 0

i=1 k = 1 , 2 , 3 . . . .

calculated at t = 0. In the special Toda shock case, they are given by (A.1.12) and (A. l .13) . We also note that, as in theorem A.I .1 , z 0 is the image of - a - 1 under z +.

+ ~ + (B+~) -2 = 1 + W2n q- W]+2n Unj, (A.1.29)

/ffil

+

where w k = p k + ~'k, and p , is defined by (A.1 .27) which is equivalent to (A.1.20) . This completes the proof of theorem A.I .1 .

It remains to solve the Marchenko equation and thus use (A.1.29) to find an expression for b n. This is done in A.2 of [13] using Fredholm determinants. The proof goes through in our case as well, with slight changes in the definitions

+ + of p k and ~', which do not affect the argument. We only present the final result.

Theorem A. 1.2. (The Dyson formula cf. [4])

b,(t)z D(n + 1, t) D(n, t) = D(n, t) 2 '

where

o l f f D(n, t) = 1 + k~=l= -~. "" d v ( z l ) ' " d v ( z , )

II~<j ( z , - zj) 2 ×

and

d z q do(z, t) = R+(z) exp[(z - z-1)t] ~ Izl=l-~

z - 1 - z i T_(z)l + I[z - z_(z)

× exp[(z - z-1)t] ~ l-,+~,zol "

(A.1.30)

In (A .1 .30)R+(z ) and T_(z) are meant to be

Remark. In section 1.3 we assume that we can take e = 0 in (A.1.30) , as in [13]. Since the second term in the RHS of (A.1.30) is bounded near - 1 , the meaning of our assumption is that we can take R+ to be zero not only on the unitary circle, but also on the circle of radius 1 - - E .

References

[1] L. Bers, Riemann Surfaces, Courant Institute Notes. [2] P.F. Byrd, M.D. Friedman, Handbook of Elliptic Inte-

grals for Engineers and Scientists, Springer, 1954. [3] E. Date, S. Tanaka, Periodic Multi-Soliton Solutions of

KdV Equation and Toda Lattice, Progr. Theor. Physics Suppl. 53, p. 107 (1976).

[4] F. Dyson, Old and New Approaches to the Inverse Scattering Problem, Essays in Honor of Valentine Barg- mann, eds. E.H. Lieb, B. Simon and A.S. Wightman, Princeton, Studies in Math. Physics (1976).

[5] H. Flaschka, On the Toda Lattice, I, Physical Review 9, 1974, 1924-5: II, Progress of Theoretical Physics, v. 51 n. 3, p. 703 (1974).

[6] B.L. Holian, H. Flaschka, D.E. McLaughlin, Shock Waves in the Toda Lattice: Analysis Phys. Rev. A24 (1981) pp. 2595-2623.

[7] B.L. Holian, G.K. Straub, Physical Review B 18, 1593 (1978).

[8] P. Koosis, Introduction to H p spaces, LMS Lecture Notes 40, 1980.

[9] P.D. Lax, C.D. Levermore, The Small Dispersion Limit of the KdV Equation, I, Communications in Pure and Applied Mathematics, v. 36, pp. 253-290.

[10] H. McKean, Theta Functions, Solitons and Singular Curves, Proceedings of the Park City Conference on Partial Differential Equations and Geometry, ed. Christopher Byrnes, Marcel Dekker 1979.

[11] R. Oba, Doubly Infinite Toda Lattice with Antisymmet- ric Asymptotics, Ph.D. Thesis, Courant Institute (1988).

[12] M. Toda, Theory of Nonlinear Lattices, Springer, 1980. [13] S. Venakides, P. Deift, R. Oba, The Toda Shock Prob-

lem, to appear in Communcations in Pure and Applied Mathematics.

[14] S. Venakides, Higher Order Lax-Levermore Theory, I, Communications in Pure and Applied Mathematics, v. 43, pp. 335-362.

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On the Toda shock problem