Multi-solitons solution of KdV equation

The general formula for the muti-soltion solution of the KdV equation is given by [2]

$$u(x,t) = 2 \frac {\partial^2}{\partial x^2} \ln {\rm det} M ,$$

where $M$ is a matrix built out of the discrete scattering data. This determinant is in the form of a sum of exponentials.

 

Figure:

[2] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. 149, CUP, Cambridge 1991.