The general formula for the mutisoltion solution of the KdV equation
is given by [2]
where 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.