Painlev\'e Analysis and Higher-Order Rogue Waves of a Generalized(3+1)-dimensional Shallow Water Wave Equation

TL;DR

This paper analyzes a (3+1)-dimensional shallow water wave equation, demonstrating its integrability, constructing higher-order rogue wave solutions, and revealing complex pattern formations with controllable geometrical structures.

Contribution

It provides the first explicit higher-order rogue wave solutions for a generalized (3+1)-dimensional shallow water wave equation using Painlevé analysis and Hirota's bilinear method.

Findings

Existence of singly-localized line-rogue waves.

Presence of doubly-localized rogue waves with multiple structures.

Ability to control pattern formations via parameter tuning.

Abstract

Considering the importance of ever-increasing interest in exploring localized waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito equation describing the unidirectional propagation of shallow-water waves and perform Painlev\'e analysis to understand its integrability nature. We construct the explicit form of higher-order rogue wave solutions by adopting Hirota's bilinearization and generalized polynomial functions. Further, we explore their dynamics in detail, depicting different pattern formation that reveal potential advantages with available arbitrary constants in their manipulation mechanism. Particularly, we demonstrate the existence of singly-localized line-rogue waves and doubly-localized rogue waves with multiple (single, triple, and sextuple) structures generating triangular and pentagon type geometrical patterns with controllable orientations that can be altered appropriately by tuning the parameters. The presented analysis will be an essential inclusion in the context of rogue waves in higher-dimensional systems.