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

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

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