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

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.

arXiv comment29 pages

References in corpus (3)

• Stability of Periodic Soliton Equations under Short Range Perturbations
• Inverse Scattering Transform for the Toda Hierarchy with Quasi-Periodic Background
• Inverse Scattering Theory for One-Dimensional Schroedinger Operators with Steplike Periodic Potentials

Cited by in corpus (15)

• Long-Time Asymptotics for the Korteweg-de Vries Equation with Steplike Initial Data
• Rarefaction Waves of the Korteweg-de Vries Equation via Nonlinear Steepest Descent
• Shock problem for MKdV equation: Long time Dynamics of the Step-like initial data
• The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step like initial profile decaying on the right half line
• A numerical dressing method for the nonlinear superposition of solutions of the KdV equation
• Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds
• Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions
• The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles
• Global well-posedness for $H^{-1}(\mathbb{R})$ perturbations of KdV with exotic spatial asymptotics
• KdV on an incoming tide
• Dispersive and soliton perturbations of finite-genus solutions of the KdV equation: computational results
• Scattering theory for Schrödinger operators on steplike, almost periodic infinite-gap backgrounds
• Low-energy expansion formula for one-dimensional Fokker-Planck and Schrödinger equations with periodic potentials
• The Transformation Operator for One-Dimensional Schroedinger Operators on Almost Periodic Infinite-Gap Backgrounds
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions