Introduction

The modified Korteveg-de Vries equation (mKdV)

               2        3
∂tu  = -  ∂x(∂ xu +  3u  )

enjoys a multisoliton solution whose explicit form is obtained using the inverse scattering method.

A double soliton q2(x,a1,a2,c1,c2) initial data gives an explicit solution

                        2           2
u (x,t) =  q2(x, a1 +  c1t,a2 +  c2t ,c1, c2).

For t 0, the solution u(x,t) is approximately given by sums of individual solitons of mKdV:

            ∑
u (x, t) ≈       --------------cj--------------,  t  ≫  0,
                 cosh cj (x - aj -  tc2 -  θ )
           j=1,2                      j     j

where, for c2 > c1 > 0,

                    (         )
              j      c2 -  c1
θ j =  ∓ (- 1 ) log   --------   ,
                      c2 +  c1

with similar expressions when c1 > c2.

For this and the explicit formula for q2, see [4] and [7, 3].

An example with a1 = 0, a2 = -1, c1 = 4 and c2 = 11 is shown below:

We want to understand the perturbed equation

               2        3
∂tu  =  - ∂x(∂ xu +  3u  -  b(x, t)u)
(1)

The external potential, b(x,t), is supposed to be slowly varying:

b(x, t) = b0 (hx, ht),  b0 ∈  C ∞,   0 <  h ≪   1.

Alternatively, as we do in numerical experiments, we can consider c large:

     c0-
c =     ,  0 <  h ≪  1.
      h

The properties of b(x,t) in that regime come from simple rescaling (see [7, 1.4]):

b(x, t) = h - 2b1 (x,h - 2t), b1 ∈ C ∞.

Here is an example with the same initial data as above, but with

                  2
b(x, t) =  75 cos  x.

This movie is obtained using the MATLAB code described below:

Bmovie(@(x,t) 75*cos(x).^2, 0.025, [0,-1,4,11],8,0)

We would like to approximate the solution of (1) with

u (x, 0) =  q (x, a ,a  , c , c )
             2     1   2  1  2

by

q2 (x,a1 (t),a2 (t),c1(t),c2(t )),

where aj(t)’s and cj(t)’s are solutions to a system of ordinary differential equations.