Stability of the cauchy problem for the korteweg-devries equation: part 1

Abstract

This paper, which is in two parts, examines the stability of the Cauchy problem for the Korteweg—deVries (KdV) equation. Let {v_m(x)} with m = 1, 2, … be a sequence of functions with the limit function . Then this work considers under what conditions as m → ∞ the solution v_m(x,t) of the KdV equation

with the initial condition

,

has as its limiting solution the function u₀(x, t) which is the solution of the Kdv equation

with the initial condition

The five conditions to be placed on the sequence of initial functions v_m(x) in order that the result is true are to be found in Theorem 5.1. They involve conditions on , an integral involving and a Wronskian involving Jost solutions of the Schrödinger equation.

Keywords:

• Cauchy problem
• Stability
• Inverse Scattering