The modified Korteveg-de Vries equation (mKdV)

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

A double soliton q_{2}(x,a_{1},a_{2},c_{1},c_{2}) initial data gives an explicit solution

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

where, for c_{2} > c_{1} > 0,

with similar expressions when c_{1} > c_{2}.

For this and the explicit formula for q_{2}, see [4] and [7, §3].

An example with a_{1} = 0, a_{2} = -1, c_{1} = 4 and c_{2} = 11 is shown below:

We want to understand the perturbed equation

| (1) |

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

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

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

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

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

by

where a_{j}(t)’s and c_{j}(t)’s are solutions to a system of ordinary differential
equations.