On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data I. Schwartz-Type Perturbations

TL;DR

This paper addresses the solution of the Korteweg-de Vries equation's Cauchy problem with steplike Schwartz-type perturbations of finite-gap potentials, considering spectral band configurations.

Contribution

It provides a novel analysis of the Korteweg-de Vries equation for specific initial data types with spectral band conditions.

Findings

Solution established for steplike Schwartz-type perturbations

Handles cases where spectral bands coincide or are disjoint

Advances understanding of initial data effects on KdV evolution

Abstract

We solve the Cauchy problem for the Korteweg-de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finite-gap potentials under the assumption that the respective spectral bands either coincide or are disjoint.