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.


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