Lump and Soliton on Certain Spatially-varying Backgrounds for an Integrable (3+1) Dimensional Fifth-order Nonlinear Oceanic Wave Model

TL;DR

This paper investigates explicit lump and soliton solutions of an integrable (3+1)-dimensional nonlinear oceanic wave model with spatially-varying backgrounds, revealing complex wave interactions and transformations influenced by background variations.

Contribution

It constructs explicit lump and kink-soliton solutions for a higher-order (3+1)D nonlinear wave model on variable backgrounds, advancing understanding of wave dynamics in such systems.

Findings

Lump solutions exhibit localized spikes and coupled bright-dark wave structures.

Soliton solutions show kink and anti-kink patterns with controllable properties.

Periodic and interacting wave structures emerge on variable backgrounds.

Abstract

The dynamics of nonlinear waves with controllable dispersion, nonlinearities, and background continues to be an exciting line of research in recent years. In this work, we focus to investigate an integrable (3+1)-dimensional nonlinear model describing the evolution of {water waves} with higher-order temporal dispersion by characterizing the dynamics of lump and soliton waves on different spatially-varying backgrounds. We construct explicit lump wave and kink-soliton solutions for the considered model through its trilinear and bilinear equations with appropriate forms of the polynomial and exponential-type initial seed solutions, respectively. Additionally, we obtain hyperbolic and periodic waves through bilinear B\"acklund transformation. We explore their propagation and transformation dynamics with spatial backgrounds through a categorical analysis and clear graphical demonstrations to fully understand the resulting solutions. Our analysis shows that the lump solution results into a simultaneous existence of well-localized spike and declining (coupled bright-dark type wave) structures on a constant background, while the soliton solution exhibits kink and anti-kink wave patterns along different spatio-temporal domains with controllable properties. When the arbitrary spatial background is incorporated appropriately, we can witness the manifestation of soliton into periodic and interacting dynamical structures. On the other hand, the lump wave results into the coexistence, interacting wave and breather formation due to periodic and localized type spatial backgrounds. The results will be an important addition to the context of nonlinear waves with controllable backgrounds in higher-dimensional models. Further, the present study can be extended to investigate several other nonlinear systems to understand the physical insights of the variable background in their dynamics.