Invariance Analysis of the (2+1) Dimensional Long Dispersive Wave Equation

Abstract

In this paper, we bring out the Lie symmetries and associated similarity reductions of the recently proposed (2+1) dimensional long dispersive wave equation. We point out that the integrable system admits an infinite-dimensional symmetry algebra along with Kac-Moody-Virasoro-type subalgebras. We also bring out certain physically interesting solutions.

A symmetry classification for a class of (2+1)-nonlinear wave equation

“…and by substituting them into invariance condition (17), we are left with a polynomial equation involving the various derivatives of u(x, y, t) whose coefficients are certain derivatives of ξ, η, τ and ϕ. Since ξ, η, τ, ϕ only depend on x, y, t, u we can equate the individual coefficients to zero, leading to the complete set of determining equations:…”

“…In section 2, the complete set of determining equations was obtained by substituting the equations ( 18), ( 19) and ( 20) in invariance condition (17) and then in section 3, we classify the symmetries of this nonlinear wave equation by assumption two cases in (36) and (37) to consider f f u is a constant or is a smooth function with respect to u and f u = 0. The commutation relations satisfied by infinitesimal generators in two cases are given in table 1, and their invariants associated with the infinitesimal generators are obtained.…”

“…Studies have also been made for (2+1)-nonlinear wave equation with constant coefficients [17,18,19]. In the special case the (2 + 1)-dimensional nonlinear wave equation…”

Generating Lie Point Symmetry Groups of (2+1)-Dimensional Broer–Kaup Equation via a Simple Direct Method

“…The invariance properties of some of the integrable nonlinear evolution equations in (2+1)-dimensions, which are very important in mathematics and physics, have been studied through the classical Lie group approaches. For instance, the Kadomatsev-Petvishilli equation, [15] the Davey-Stewartson equation, [16] the three-wave interaction problem, [17] the cylindrical Kadomtsev-Petviashvilli equation, [18] the stimulated Raman scattering equation, [19] the Nizhnik-Novikov-Veselov equation, the breaking soliton equation, the sine-Gordon system, [20] and the long dispersive wave equation, [21,22] etc. The related symmetry groups can be obtained by means of * The project supported by National Natural Science Foundation of China under Grant No.…”