Lie symmetry analysis and exact explicit solutions of three-dimensional Kudryashov–Sinelshchikov equation

Lie Symmetry Analysis and New Exact Solutions for a Higher-Dimensional Shallow Water Wave Equation

“…The symmetry group of (1) will be generated by the vector field of the form (3). Applying the fourth prolongation pr (5) to (1), we find that the coefficient functions , , , , and must satisfy the symmetry condition…”

“…where 3 = , , , , and are the total derivatives with respect to , , , and , respectively. Substituting (5) into (4), combined with (1) and equating the coefficients of the various monomials in the first, second, third, and the other partial derivatives and various powers of , we can find the determining equations for the symmetry group of (1); then standard symmetry group calculations lead to the following forms of the coefficient functions:…”

“…However, the study on nonlinear wave solution is few and there is no unified approach. In this work, we studied the nonlinear wave solution of a higher-dimensional shallow water wave equation by using Lie symmetry analysis [1][2][3][4][5] and extend F-expansion method [6][7][8][9].…”

New exact solutions for Kudryashov–Sinelshchikov equation

“…In [14], the author obtained some soliton solutions to the nonlinear (3+1)-dimensional variable-coefficient Kudryashov-Sinelshchikov model by using an auto-Bäcklund transformation. In [15], the authors obtained all of the geometric vector fields of the equation and some new exact explicit solutions to the 3-dimensional Kudryashov-Sinelshchikov equation by using the Lie symmetry analysis. In [16], the authors applied the Lie group method to derive the symmetries of the Kudryashov-Sinelshchikov equation.…”

Symmetry Reduction and Numerical Solution of Von K a ´ rm a ´ n Swirling Viscous Flow

“…There have been several studies about the symmetry method, such as symmetry classification [8], potential symmetry [9], approximate symmetry [10], etc. Based on the symmetries of a PDE, many important properties of the equation such as Lie algebras [11,12], conservation laws [13][14][15][16][17][18], and exact solutions [16][17][18][19][20][21][22] can be considered successively. Recently, some researchers focus on the applications of the symmetry method for solving boundary value problems (BVP) of a PDE [2,[23][24][25].…”