On the Existence of Self-Similar solutions for some Nonlinear Schr\"odinger equations

TL;DR

This paper constructs and analyzes stable, global, self-similar solutions to nonlinear Schrödinger equations that exhibit weak localization and deviate from standard wave asymptotics, including scenarios with two bubbles.

Contribution

It introduces the existence of stable, asymptotically self-similar solutions with non-zero L^2 norms for certain nonlinear Schrödinger equations, expanding understanding of their long-term behavior.

Findings

Solutions are asymptotically self-similar as time approaches infinity.

Solutions include configurations with two bubbles.

Such solutions are stable and have non-zero L^2 norms.

Abstract

We construct solutions of Schr\"odinger equations which are asymptotically self-similar solutions as time goes to infinity. Also included are situations with two bubbles. These solutions are global, with non-zero $L^2$ norms, and are stable. As such they are not of the standard asymptotic decomposition of linear waves and localized waves. Such weakly localized solutions were expected in view of previous works \cite{Liu-Sof1,Liu-Sof2} on the large time behavior of general dispersive equations. It is shown that one can associate a \emph{scattering channel} to such solutions, with the dilation operator as the asymptotic ``Hamiltonian''.