Construction of a new (3 + 1)-dimensional KdV equation and its closed-form solutions with solitary wave behaviour and conserved vectors

TL;DR

This paper constructs a new (3+1)-dimensional KdV equation, finds exact solutions including solitary waves, and computes conserved vectors, advancing understanding of higher-dimensional nonlinear wave equations.

Contribution

It introduces a novel (3+1)-dimensional KdV equation, derives exact solutions using Lie symmetries, and calculates conserved vectors, which are new contributions to nonlinear wave theory.

Findings

Seven exact solutions with solitary wave behavior

Graphical illustrations of wave propagations

Conserved vectors computed via Ibragimov's method

Abstract

This paper discusses the construction of a new $(3+1)$-dimensional Korteweg-de Vries (KdV) equation. By employing the KdV's recursion operator, we extract two equations, and with elemental computation steps, the obtained result is $ 3u_{xyt}+3u_{xzt}-(u_{t}-6uu_{x}+u_{xxx})_{yz}-2\left(u_{x}\partial_{x}^{-1}u_{y}\right)_{xz}-2\left(u_{x}\partial_{x}^{-1}u_{z}\right)_{xy}=0 $. We then transform the new equation to a simpler one to avoid the appearance of the integral in the equation. Thereafter, we apply the Lie symmetry technique and gain a $7$-dimensional Lie algebra $L_7$ of point symmetries. The one-dimensional optimal system of Lie subalgebras is then computed and used in the reduction process to achieve seven exact solutions. These obtained solutions are graphically illustrated as 3D and 2D plots that show different propagations of solitary wave solutions such as breather, periodic, bell shape, and others. Finally, the conserved vectors are computed by invoking Ibragimov's method.