Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation

TL;DR

This paper analyzes the Fourier space asymptotics of self-similar solutions to the modified Korteweg-de Vries equation, providing new insights into their stability and behavior in Fourier space, especially relating to Painlevé II solutions.

Contribution

It introduces a fixed point method to determine Fourier asymptotics of self-similar solutions, linking Fourier and physical space descriptions for the first time.

Findings

Derived Fourier asymptotics for self-similar solutions.

Connected Fourier space constants with physical space constants.

Enhanced understanding of stability properties of solutions.

Abstract

We give the asymptotics of the Fourier transform of self-similar solutions to the modified Korteweg-de Vries equation, through a fixed point argument in weighted W^{1,\infty} around a carefully chosen, two term ansatz. Such knowledge is crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow. In the defocusing case, the self-similar profiles are solutions to the Painlev\'e II equation. Although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. We are able to relate the constants involved in the description in Fourier space with those involved in the description in physical space.