The \texorpdfstring{$\mathbf N$-soliton}{N-soliton} solution formula for the Kaup--Kupershmidt equation

Abstract

An explicit formula is presented for the $\mathbf N$-soliton solution to the Kaup--Kupershmidt equation, for any positive integer $\mathbf N,$ by using the input data set consisting of $\mathbf N$ distinct positive $p_j$-values and $\mathbf N$ arbitrarily chosen positive $c_j$-values. The formula is derived from the solution to the inverse scattering problem for a third-order linear operator in the reflectionless case with the $p_j$-values related to the bound-state poles of a transmission coefficient and the $c_j$-values related to the bound-state dependency constants. It is indicated how the formula is constructed from the solution to a Riemann--Hilbert problem and from the solution to a Marchenko integral equation used in the formulation of the aforementioned inverse scattering problem.