Reductions of the (4 + 1)-dimensional Fokas equation and their solutions

TL;DR

This paper introduces the (4+1)-dimensional Fokas equation as an integrable extension of KP and DS equations, constructs various exact solutions, and explores their complex dynamics and potential applications in physics.

Contribution

The paper develops determinant-based solutions for the (4+1)-dimensional Fokas equation and demonstrates its relation to lower-dimensional equations, expanding the understanding of multidimensional integrable systems.

Findings

Constructed soliton, breather, rational, and semi-rational solutions.

Identified rogue wave-like patterns in high-order lumps.

Presented new semi-rational solutions and classified lump fission and fusion processes.

Abstract

An integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations is investigated in this paper.We will refer to this integrable extension as the (4+1)-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the (4 + 1)-dimensional Fokas equation are constructed based on the Hirota's bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained high-order lumps are similar to those of rogue waves in the (1 + 1)-dimensions by choosing different values of the free parameters of the model. Furthermore, three kinds of new semi-rational solutions are presented and the classification of lump fission and fusion processes is also discussed. Additionally, we give a new way to obtain rational and semi-rational solutions of (3 + 1)-dimensional KP equation by reducing the solutions of the (4 + 1)-dimensional Fokas equation. All these results show that the (4 + 1)-dimensional Fokas equation is a meaningful multidimensional extension of the KP and DS equations. The obtained results might be useful in diverse fields such as hydrodynamics, non-linear optics and photonics, ion-acoustic waves in plasmas, matter waves in Bose-Einstein condensates, and sound waves in ferromagnetic media.