A universal asymptotic regime in the hyperbolic nonlinear Schr\"odinger equation

TL;DR

This paper identifies a universal long-time asymptotic regime in the hyperbolic nonlinear Schrödinger equation, demonstrating that diverse initial conditions evolve towards a self-similar attractor, highlighting its robustness and universality.

Contribution

The study introduces an approximate self-similar solution as a universal attractor for the hyperbolic nonlinear Schrödinger equation across various initial conditions.

Findings

Existence of a universal asymptotic regime

Self-similar solution approximates long-term behavior

Robustness across different initial amplitudes

Abstract

The appearance of a fundamental long-time asymptotic regime in the two space one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is discussed. Based on analytical and extensive numerical simulations an approximate self-similar solution is found for a wide range of initial conditions -- essentially for initial lumps of small to moderate energy. Even relatively large initial amplitudes, which imply strong nonlinear effects, eventually lead to local structures resembling those of the self-similar solution, with appropriate small modifications. These modifications are important in order to properly capture the behavior of the phase of the solution. This solution has aspects that suggest it is a universal attractor emanating from wide ranges of initial data.