Small-amplitude self-similar solutions for one-dimensional nonlinear dispersive equations

TL;DR

This paper develops a systematic method for constructing small-amplitude self-similar solutions for one-dimensional nonlinear dispersive equations, providing detailed asymptotic analysis across different frequency scales.

Contribution

It introduces a new approach for constructing and analyzing self-similar solutions in nonlinear dispersive equations, applied to three classical models.

Findings

Constructed small-amplitude self-similar solutions for selected models

Provided asymptotic descriptions at small and large frequencies

Demonstrated the applicability of the method to multiple dispersive equations

Abstract

Given a nonlinear dispersive equation which admits a scaling invariance, there may exist self-similar solutions. In this work, we present a systematic approach for the construction of small-amplitude self-similar solutions, together with precise asymptotic descriptions at both small and large frequency scales. These ideas are then applied to three classic dispersive models: the modified Benjamin-Ono, the quartic Korteweg-de Vries and the cubic nonlinear Schr\"odinger equations.