Some new periodic solitary wave solutions of (3+1)-dimensional generalized shallow water wave equation by Lie symmetry approach

A new implementation of a novel analytical method for finding the analytical solutions of the (2+1)-dimensional KP-BBM equation

“…There are several methods to solve and find the exact solutions to the evolution equations involving non-linearity. For example, the -expansion method [6] , [7] , [8] , [9] , the exp-function method [10] , the modified exp-function method [11] , [12] , the tanh–coth expansion method [13] , [14] , [15] , the improved method [16] , [17] , the -expansion method [18] , [19] , [20] , [21] , the simple equation method (SEM) [22] , [23] , the Lie symmetry approach [24] , [25] , the sine-Gordon method [26] , the modified Sardar sub-equation method [27] , [28] , the generalized Kudryashov method [29] , [30] , the Riccati-Bernoulli sub-ODE method [31] , [32] , the improved generalized Riccati mapping method [33] , the modified double sub-equation method [34] , the generalized exponential rational function (GERF) method [35] , [36] and there are many more. Beside these integer order PDEs there are lots of techniques for investigating fractional order PDEs such as [37] , [38] etc.…”

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

“…In [21], Huang studied the stability of solitons by numerical methods and noticed that the soliton amplitude magnitude is affected by the spectral parameters. In [22], the closed-form solutions of the equation were derived by Lie symmetry, and the soliton solutions were found through the optimal system. Based on the auto-Bäcklund transformation, Li and Liu [23] constructed the multi-periodic solitons of Eq.…”

Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

“…New analytic solutions, including periodic solitary wave, cross-kink soliton and doubly periodic breather-type solutions, were obtained for a type of (3 + 1)dimensional shallow water wave equation by using an optimal system of Lie symmetry vectors. [32] In Ref. [33], Hirota's bilinear method was used to determine the multiple-soliton solutions of the (3 + 1)-dimensional shallow water wave equations (1) and ( 2), which also testified the complete integrability of the two equations.…”