Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D

TL;DR

This paper introduces a new coordinate system to analyze the hyperbolic cubic nonlinear Schrödinger equation in 2D, focusing on the existence of special solutions like standing waves and self-similar waves.

Contribution

It develops a novel coordinate framework enabling the study of hyperbolically radial solutions for the 2D hyperbolic cubic NLS, including standing and self-similar waves.

Findings

Existence of bounded hyperbolically radial standing waves

Existence of hyperbolically radial self-similar solutions

Method adaptable to more general nonlinearities

Abstract

In this note we propose a new set of coordinates to study the hyperbolic or non-elliptic cubic nonlinear Schrodinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.