The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles

Abstract

We develop the inverse scattering transform for the KdV equation with real singular initial data $q(x)$ of the form $q(x)=r^{\prime }(x)+r(x{)}^2$, where $r\in L_{\text{loc}}^2$ and $r=0$ on ${\mathbb{R}}_{+}$. As a consequence we show that the solution $q(x,t)$ is a meromorphic function with no real poles for any $t>0$.

arXiv admin note: text overlap with arXiv:1310.3873

References in corpus (4)

• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials
• On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data I. Schwartz-Type Perturbations
• The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step like initial profile decaying on the right half line
• Generalized reflection coefficients

Cited by in corpus (4)

• On classical solutions of the KdV equation
• KdV equation beyond standard assumptions on initial data
• Global well-posedness for $H^{-1}(\mathbb{R})$ perturbations of KdV with exotic spatial asymptotics
• KdV on an incoming tide