Asymptotic decay of solitary wave solutions of the fractional nonlinear Schr\"{o}dinger equation

TL;DR

This paper investigates the long-range decay behavior of solitary wave solutions to the one-dimensional fractional nonlinear Schrödinger equation, revealing algebraic decay rates influenced by the fractional order, supported by numerical experiments.

Contribution

It provides a detailed analysis of the asymptotic decay rates of solitary waves in the fractional NLS, extending previous existence results with new decay characterizations.

Findings

Solitary waves decay algebraically at infinity.

Decay rate depends on the fractional order parameter.

Numerical experiments confirm the theoretical decay rates.

Abstract

The existence of solitary wave solutions of the one-dimensional version of the fractional nonlinear Schr\"{o}dinger (fNLS) equation was analyzed by the authors in a previous work. In this paper, the asymptotic decay of the solitary waves is analyzed. From the formulation of the differential system for the wave profiles as a convolution, these are shown to decay algebraically to zero at infinity, with an order which depends on the parameter determining the fractional order of the equation. Some numerical experiments illustrate the result.