Families of rational and semi-rational solutions of the partial reverse space-time nonlocal Mel'nikov equation

TL;DR

This paper introduces a nonlocal Mel'nikov equation, derives various solutions including rational and semi-rational types, and explores their connections to rogue waves and other phenomena using Hirota's method.

Contribution

It presents the first derivation of rational and semi-rational solutions for the partial reverse space-time nonlocal Mel'nikov equation using Hirota's bilinear method.

Findings

Derived soliton, breather, and mixed solutions.

Obtained rational (lump) and semi-rational solutions from long wave limits.

Generated rogue waves and related solutions under specific parameters.

Abstract

Inspired by the works of Ablowitz, Mussliman and Fokas, a partial reverse space-time nonlocal Mel'nikov equation is introduced. This equation provides two dimensional analogues of the nonlocal Schrodinger-Boussinesq equation. By employing the Hirota's bilinear method, soliton, breathers and mixed solutions consisting of breathers and periodic line waves are obtained. Further, taking a long wave limit of these obtained soliton solutions, rational and semi-rational solutions of the nonlocal Mel'nikov equation are derived. The rational solutions are lumps. The semi-rational solutions are mixed solutions consisting of lumps, breathers and periodic line waves. Under proper parameter constraints, fundamental rogue waves and a semi-rational solutions of the nonlocal Schrodinger-Boussinesq equation are generated from solutions of the nonlocal Mel'nikov equation.