Characteristic function method for classification of equations of hydrodynamics of a perfect fluid

Non-Traveling Wave Solutions for the (1 + 1)-Dimensional Burgers System by Riccati Equation Mapping Approach

“…In the past few decades, many significant methods have been presented such as Bäklund transformation, Darboux transformation, the extended tanh-function method, and the F-expansion method, Lie group analysis, homogeneous balance method, Jacobi elliptic function method, and the mapping method, etc. [9][10][11][12][13][14][15]. The mapping approach is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations, the remarkable characteristic of which is that we can have many different ansatzs and therefore, a large number of solutions.…”

Traveling Wave Solutions and Kind Wave Excitations for the (2 + 1)-Dimensional Dissipative Zabolotskaya-Khokhlov Equation

“…So it is always an interesting topic to search for meaningful solutions for PDEs. In order to find some new exact solutions, a wealth of effective methods have been set up, for instance, the bilinear method, the standard Painlevé truncated expansion, the method of "coalescence of eigenvalue" or "wavenumbers", the homogenous balance method, the homotopy-perturbation method, the hyperbolic function method, the Jacobian elliptic method, the (G'/G)-expansion method, the variable separation method, and the mapping equation method [6][7][8][9][10][11][12][13][14][15], etc. Among these methods, the mapping equation approach is one of the most effectively straightforward algebraic methods to construct exact solutions of NPDE [16][17][18][19].…”

New Mapping Solutions and Line Soliton Excitations in a (1+1)-Dimensional Dispersive Long-Water Wave System

“…In the study of nonlinear physics and mathematics, the search of new exact solutions of nonlinear evolution equations (NEEs) is one of the most important problems. Various methods for obtaining exact solutions to NEEs have been proposed,such as a bilinear method, symmetry reductions, standard and extended Painleve truncated expansions,the method of "coalescence of eigenvalue" or "wavenumbers", the homogenous balance method, the "variable separation approach", Lie group method of infinitesimal transformations, the nonclassical Lie group method, Clarkson and Kruskal direct method (CK), conditional similarity reduction method [1][2][3][4] and the mapping approach, [5][6][7][8][9] etc.…”