The focusing logarithmic Schr\"odinger equation: analysis of breathers and nonlinear superposition

TL;DR

This paper analyzes the focusing logarithmic Schrödinger equation, demonstrating stability of Gaussian solutions, periodic behavior in one dimension, and establishing a nonlinear superposition principle for multiple Gaussian initial data.

Contribution

It introduces a nonlinear superposition principle for Gaussian solutions and analyzes their stability and periodicity in the focusing logarithmic Schrödinger equation.

Findings

Gaussian initial data remains Gaussian over time

Gausson is an orbitally stable solution

Solution with multiple Gaussian initial data stays close to the sum of individual solutions for long times

Abstract

We consider the logarithmic Schr\"odinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson-a time-independent Gaussian function-is an orbitally stable solution. In the general case in dimension $d = 1$, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in $L^2$) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions.