Nonlinear Equations for Soliton Eigenfunctions Are the IST Integrable Equations

Abstract

A starting and basic point of the inverse spectral transform (IST) method is the representation of the given nonlinear partial differential equation as the compatibility condition of some nontrivial linear system

$$\begin{array}{*{20}{c}} {{{L}_{1}}(U;\lambda )\Psi } \hfill & = \hfill & {0,} \hfill \\ {{{L}_{2}}(U;\lambda )\Psi } \hfill & = \hfill & 0 \hfill \\ \end{array}$$

(1)

where L ₁ and L ₂ are usually differential operators the coefficients of which depend on the field U(x,t), U _x , U _xx ,... and spectral parameter λ (see e.g. [1–4]). Integrable equation for U arises after the elimination of the eigenfunction Ψ from the system (1) [1–4].