Self-similar solutions to the derivative nonlinear Schr\"odinger equation

TL;DR

This paper investigates a class of self-similar solutions to the derivative nonlinear Schrödinger equation, revealing a unique logarithmic phase correction arising from nonlinear interactions, contrasting with previous linear-based findings.

Contribution

It demonstrates that the logarithmic phase correction in these solutions results from nonlinear interactions, unlike the linear origin in pseudo-conformally invariant cases.

Findings

Logarithmic phase correction from nonlinear interactions

Distinct behavior compared to pseudo-conformally invariant case

Asymptotic analysis of profile functions

Abstract

A class of self-similar solutions to the derivative nonlinear Schr\"odinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions.