A new asymptotic regime for the KdV equation with Wigner-von Neumann type initial data

TL;DR

This paper studies the long-time behavior of solutions to the KdV equation with specific initial data causing Wigner-von Neumann resonances, revealing a new resonance regime with slower decay rates.

Contribution

It introduces a novel asymptotic regime for the KdV equation linked to Wigner-von Neumann resonances, extending understanding of long-range effects in integrable systems.

Findings

Wigner-von Neumann resonances induce a distinct asymptotic regime.

Resonance regime exhibits slower decay compared to standard radiant waves.

Adapted inverse scattering transform handles potentials with spectral singularities.

Abstract

We investigate the long-time asymptotic behavior of solutions to the Cauchy problem for the KdV equation, focusing on the evolution of the radiant wave associated with a Wigner-von Neumann (WvN) resonance induced by the initial data (potential). A WvN resonance refers to an energy level where the potential exhibits zero transmission (complete reflection). The corresponding Jost solution at such energy becomes singular, and in the NLS context, this is referred to as a spectral singularity. A WvN resonance represents a long-range phenomenon, often introducing significant challenges, such as an infinite negative spectrum, when employing the inverse scattering transform (IST). To avoid some of these issues, we consider a restricted class of initial data that generates a WvN resonance but for which the IST framework can be suitably adapted. For this class of potentials, we demonstrate that each WvN resonance produces a distinct asymptotic regime -- termed the resonance regime -- characterized by a slower decay rate for large time compared to the radiant waves associated with short-range initial data.