Reply to the comment by M A Almeidaet al

Traveling-Wave Solutions of the Calogero-Degasperis-Fokas Equation in 2+1 Dimensions

“…For b = 0 and λ = 0, Eq. (8) admits symmetries (13). For λ = 0, in addition to the previous ones, it admits the generator…”

“…In most cases, the corresponding Lie algebra has a Kac-Moody-Virasoro-type subalgebra, but some integrable (2+1)-dimensional equations do not admit a Virasoro-type subalgebra. Examples of such equations are a breaking soliton equation introduced by Bogoyavlenskii, a (2+1)-dimensional generalization of the nonlinear Schrödinger equation [13], and the SKdV equation [14].…”

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

“…is a smooth function on u, by using here the classical Lie symmetric method. 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…”

Lie symmetries and singularity analysis for generalized shallow-water equations

“…The are many applications of the Lie symmetries on the analysis of differential equations, for the determination of exact solutions, to determine conservation laws, study the integrability of dynamical systems or classify algebraic equivalent systems [6][7][8][9][10][11][12][13]. Integrability is a very important property of dynamical systems, hence it worth to investigate if a given dynamical system is integrable [14][15][16][17][18][19][20][21].…”