Periodic Solutions and Rogue Wave Type Extended Compactons in the Nonlinear Schrodinger Equation and \phi^{4} Theories

TL;DR

This paper investigates special solutions to the nonlinear Schrödinger equation and ^4 theories, proposing a method to generate rogue wave-like solutions that extend over space, and analyzes their evolution into steady states.

Contribution

It introduces a novel approach to construct rogue wave-like solutions in NLSE and ^4 theories using a mapping to anharmonic oscillators and solution truncation techniques.

Findings

Proposes a method to create approximate rogue wave solutions by truncating and stitching solutions.

Shows rogue wave solutions can evolve into steady states in ^4 theory.

Identifies that rogue waves can appear spontaneously due to time-reversal symmetry.

Abstract

By employing a mapping to classical anharmonic oscillators, we explore a class of solutions to the Nonlinear Schrodinger Equation (NLSE) in 1+1 dimensions and, by extension, asymptotically in general dimensions. We discuss a possible way for creating approximate rogue wave like solutions to the NLSE by truncating exact solutions at their nodes and stitching them with other solutions to the NLSE. The resulting waves are similar to compactons with the notable difference that they are not localized but rather extend over all of space. We discuss rogue waves in a \phi^4 field theory in the context of a discretized Lagrangian and rogue wave behavior is shown to evolve into a steady state. Due to time-reversal invariance of this theory, the steady state found could alternatively evolve into a rogue wave giving rise to a large wave which seems to appear from nothing.