The solutions of classical and nonlocal nonlinear Schr\"{o}dinger equations with nonzero backgrounds: Bilinearisation and reduction approach

TL;DR

This paper introduces a bilinearisation-reduction method to derive explicit solutions, including rogue waves, for classical and nonlocal nonlinear Schrödinger equations with nonzero backgrounds.

Contribution

It develops a unified approach using Wronskians to solve both classical and nonlocal NLS equations with nonzero backgrounds, including explicit rogue wave solutions.

Findings

Derived solutions in terms of quasi double Wronskians.

Obtained explicit rogue wave formulas of arbitrary order.

Analyzed solutions for focusing and reverse-space nonlocal NLS equations.

Abstract

In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schr\"{o}dinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a pair of solutions $(q_0,r_0)$ we bilinearize the unreduced system and obtain solutions in terms of quasi double Wronskians. Then we implement reductions by introducing constraints on the column vectors of the Wronskians and finally obtain solutions to the reduced equations, including the classical NLS equation and the nonlocal NLS equations with reverse-space, reverse-time and reverse-space-time, respectively. With a set of plane wave solution $(q_0,r_0)$ as a background solution, we present explicit formulae for these column vectors. As examples, we analyze and illustrate solutions to the focusing NLS equation and the reverse-space nonlocal NLS equation. In particular, we present formulae for the rouge waves of arbitrary order for the focusing NLS equation.