Classical Lie symmetries and similarity reductions of the (2 + 1)-dimensional dispersive long wave system

Abstract

In the present work, classical Lie symmetry method is adopted to obtain the Lie symmetries and similarity reductions of the (2 + 1)-dimensional dispersive long wave equation. We also demonstrate that the symmetry algebra of this equation is an infinte dimensional, together with Kac–Moody–Virasoro type subalgebras.

Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma–Tasso–Olver Equation

“…e motive of this present section is to describe the main steps used in obtaining the infinitesimal, Lie symmetries, and optimal system of one-dimensional subalgebra for STO equation (2). To find the invariant solutions of (2), the STM is applied.…”

“…where the infinitesimals ξ 1 � ξ (1) , ξ 2 � ξ (2) , ξ 3 � τ, and η are the infinitesimals transformations for x, y, t, and u, respectively. e linked vector field V with the above group of transformation is defined as…”

“…where c i ; 1 ≤ i ≤ 4 are arbitrary constants and f 1 (t) and f 2 (t) are functions obtained in the processing of infinitesimals. e choice of f i ′ s can achieve some physical structures of solutions of equation (2). e symmetry algebra of equation ( 2) can be obtained by the given vectors…”

“…Insert the above value of u(x, y, t) from (40) into STO equation (2). en, the reduced equation is given as [Ad]…”

“…e various types of rational solutions were investigated by analytical and numerical methods such as the Lie symmetry method [2,3], homogeneous balance method [4], tanh function method [5], Hirota bilinear method [6], Backlund transformation method [7], Darboux transformation method [8], the truncated Painleve expansion [9], and F-expansion method [10]. In solitary wave theory, a large variety of traveling waves are of much concern.…”