N Soliton Solutions to The Bogoyavlenskii-Schiff Equation and A Quest for The Soliton Solution in (3 + 1) Dimensions

TL;DR

This paper explores soliton solutions to the Bogoyavlenskii-Schiff equation in higher dimensions, providing explicit multi-soliton solutions and proposing a new (3+1)-dimensional equation guided by symmetry considerations.

Contribution

It introduces a new approach to integrable systems in higher dimensions using trilinear forms and proposes a novel (3+1)-dimensional equation based on symmetry.

Findings

Explicit multi-soliton solutions for the (2+1)-dimensional BS equation.

Identification of a new feature in the soliton solutions.

Proposal of a new (3+1)-dimensional integrable equation.

Abstract

We study the integrable systems in higher dimensions which can be written not by the Hirota's bilinear form but by the trilinear form. We explicitly discuss about the Bogoyavlenskii-Schiff(BS) equation in (2 + 1) dimensions. Its analytical proof of multi soliton solution and a new feature are given. Being guided by the strong symmetry, we also propose a new equation in (3 + 1) dimensions.