AbstractIn a recent paper, Jin, Levermore and McLaughlin analyze the semiclassical behavior of solutions to the defocusing, completely integrable nonlinear Schrödinger equation. We complete their analysis, by providing the long time behavior of the semiclassical solutions

Kamvissis, S.

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PERGAMON Applied Mathematics Letters 12 (1999) 35-37 
www.elsevier, nl/locate/aml 
Long T ime Behavior for Semiclassical NLS 
S. KAMVISSIS 
Max-Planck-Institute for Mathematics in the Sciences 
Leipzig, Germany 
and 
University of Aix-MarseiUes, France 
(Received March 1999; accepted April 1999) 
Communicated by C. Bardos 
Abst rac t - - In  a recent paper, Jin, Levermore and McLaughlin analyze the semiclassical behavior 
of solutions to the defocusing, completely integrable nonlinear SchrSdinger equation. We complete 
their analysis, by providing the long time behavior of the semiclassical solutions. © 1999 Elsevier 
Science Ltd. All rights reserved. 
Keywords--Sol itons, Nonlinear SchrSdinger quation, Semiclassical symptotics. 
In their well-known papers Lax and Levermore [1] analyze the solutions of the zero dispersion 
limit of KdV, under fairly general initial data, either belonging in the Schwartz class, or of shock 
type. In the third of the series, they also give the long-time asymptotics for such solutions. 
Following Lax and Levermore, an analogous discussion of the semiclassical defocusing NLS 
equation is given in [2]. Whitham equations are introduced and weak limits of the squared 
density, the momentum, and the energy of solutions are expressed in terms of the Riemann 
invariants of the Whitham system. Although long-time formulae are not given in [2], they would 
be of some value 1. The statement and proof of such formulae is the aim of this note. 
THEOREM. Let u(x,t; h) solve 
h 2 
ihut(x, t; h) + -~uxx(x, t; h) + (1 - lu(x, t; h)] 2) u(x, t; h) = O, 
with the far  - field boundary condition 
u(x , t )~expk- -h - -  ) , for some S~ • S, 
% /  
and the initial condition 
(1) 
1They could be used, for example, in evaluating the k - e turbulence model for the Navier-Stokes equations [3]. 
0893-9659/99/$ - see front matter. (~) 1999 Elsevier Science Ltd. All rights reserved. Typeset by .A.h/$S-~]~/X 
PII: S0893-9659(99)00118-4 
36 S. KAMVISSIS 
nlln 
ma~ 
Figure 1. The initial data r~=(x). Note their critical values )~min and Amax, and the 
indicated defining relations for the turning points x q- (A). 
where A(x) - I and the x-derivative Sz belong in the Schwartz class. Let us also assume, for 
simplicity, that the initial data are 'single well' in the sense of [2] (see Figure 1). In other words, 
the r defined below has only one maximum = Amax (respectively, r+ has only one minimum 
= Amin) and -1  < Amax < Amin < 1. Then, the weak limit ~(x,t) = limh--.o [u(x,t;h)[ 2 exists 
and in the 'Whitham' region x / t  E (-1,Amax) U (Amin, 1), we have, as t ~ 0% 
f i (x , t )~ ' , l -4¢( t )  1-  where 
f~+(~) [A-  1/2(r+(s) +r_ (s ) )  I
sx + A(x), r±(x) = -y  
and x± are defined by r_(x±(A)) = A, x_ < x+. 
(2) 
Outside the Whitham region, ~ ~ 1. 
PROOF. The existence of the weak limit is proved in [2]. The long-time behavior can be derived 
following [1] in two ways. One can use the semiclassical formulae to derive long-time asymptotics 
for the Riemann invariants of the Whitham equations. We prefer to follow an alternative way 
(also suggested in [1]) of beginning with the multisoliton formula for fixed h and then taking 
h~0.  
For fixed h, the long-time behavior of ]ut 2 is as follows [4, pp. 168-176]. In the solitonless 
regions [x/t[ > 1 and Am~x < x/ t  < Amin, we have ]ul 2 = 1 -O( t -1 /2 ) ,  as t --* co. In the 
Whitham region, the solution is a multisoliton solution: 
N(h) 
lu(x,t; hi[ 2 ~ i - ~ s(x - ~t - x=,~=), 
n=l  
1 - 7/2 
s(x, 7) = cosh2((1 _ ~2)l/2(x/2h) ) ,
where 
(3) 
with exponentially small error. The eigenvalues of the associated Lax operator ~/n accumulate in 
the set ( -1 ,  Amax) U (Amin, 1). The xns are some phase constants of no importance. 
Long Time Behavior 37 
The width  of each sol iton s(x, zl) is O(h/(1 - zl2)1/2). By Weyl 's  Law for the d ist r ibut ion of 
eigenvalues in ( -1 ,  Amax) t3 (Amin, 1) as h --* 0, 
~h 
~+1 - v~ ¢(#~), (4) 
where ~n E (~?n,~n+l). 
Peaks of sol itons are located at ~nt. As t --~ oc, they are separated by 7rht/¢(~n), so for large t, 
they are well separated.  The wave number 77 of the soliton that  peaks at x at t ime t is ~ == x/t,  
if t is large and either -1  < x/t  < Amax or A,,in < x/t  < 1. So the density of the solitons is 
¢(x/t)  (5) 
7rht 
The area between a soliton and the line u = 1 is 
4h(1-n  2) 1/2~4h 1-  -/ , (6) 
so the asymptot ic  area density is the product  of (5) and (6): 
The asymptot ic  area density is 1 - p. Hence, the asymptot ic  formula for the weak l imit t5 follows 
readily. 
REFERENCES 
1. P.D. Lax and C.D. Levermore, The zero dispersion limit for the KdV equation, I II, III, Comm. Pur~; Appl. 
Math. 36, 253-290, 571-593, 809-830, (June 1983). 
2. S. Jin, C.D. Levermore and D.W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, 
Comm. Pure Appl. Math. (to appear); The Behavior of Solutions of the NLS Equation in the Semiclassical 
Limit, In Singular Limits of Dispersive Waves, (Edited by N.M. Ercolani et al.) (NATO ASI Series), Plenum 
Press, New York, (1994). 
3. C. Bardos, private communication. 
4. L.D. Fadeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, (1987). 


Long time behavior for semiclassical NLS 

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Long time behavior for semiclassical NLS

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