’’IST-solvable’’ nonlinear evolution equations and existence—An extension of Lax’s method

Abstract

We present here a new and easy method, a natural extension of Lax’s method, for obtaining general ’’IST-solvable’’ nonlinear evolution equations. These are evolution equations for the potential function(s), v, of a Hamiltonian, H, when the logarithmic t derivatives of H’s inverse scattering data are given by a t-dependent ratio of entire functions of E, Ω (t,E). Here E is the energy variable and Ω is the ’’dispersion relation’’ of Abowitz, Kaup, Newell, and Segur (AKNS). We pose the question of existence of the evolution equation’s solution. This question is answered completely in the one-dimensional Schrödinger case (first example). In a second example we derive the evolution equation for an n×n matrix generalization of the Zakharov–Shabat–AKNS equation. Our method displays the central role of analyticity in E in the IST method as a whole.