Classical and Nonclassical symmetries of the (2+1)-dimensional Kuramoto-Sivashinsky equation

TL;DR

This paper analyzes the symmetries of the (2+1)-dimensional Kuramoto-Sivashinsky equation, identifying classical symmetries and their algebraic structure, while showing the absence of nonclassical symmetries.

Contribution

It provides a comprehensive symmetry analysis of the (2+1)-dimensional Kuramoto-Sivashinsky equation, including classical and nonclassical symmetries, and constructs the Lie algebra structure.

Findings

Classical symmetries and Lie algebra structure identified.

Optimal subalgebras constructed for the equation.

No nonclassical symmetries found for the model.

Abstract

In this paper, we have studied the problem of determining the largest possible set of symmetries for an important example of nonlinear dynamical system: the Kuramoto-Sivashinsky (K-S) model in two spatial and one temporal dimensions. By applying the classical symmetry method for the K-S model, we have found the classical symmetry operators. Also, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras of the equation is constructed. The Lie invariants associated to the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. By applying the nonclassical symmetry method for the K-S model we concluded that the analyzed model do not admit supplementary, nonclassical type, symmetries. Using this procedure, the classical Lie operators only were generated.