Lump solutions of a nonlinear PDE containing a third-order derivative of time

Three types of periodic solutions of new (3 + 1)‐dimensional Boiti–Leon–Manna–Pempinelli equation via bilinear neural network method

“…And then, this direct method applied to solve the solutions of many NLEEs such as complex wave solutions, 11 lump-stripe solutions, lump-kink solutions, lump-type solutions, high-order lump solutions, and their interaction solutions. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] Recently, Zhang and Bilige 28 combined the shallow neural network model with the bilinear method called bilinear neural network method (BNNM) to flexibly produce a variety of test functions, based on which more unknown and interesting solutions to NLEEs can be explored. It is worth mentioning that the BNNM framework has been able to construct lump solutions, periodic solutions, bright-dark solitons solutions, and so on.…”

“…For example, based on the Hirota bilinear method, Ma 9,10 proposed a direct method that is taking the solution of bilinear equation as a positive quadratic function to get the lump solutions. And then, this direct method applied to solve the solutions of many NLEEs such as complex wave solutions, 11 lump–stripe solutions, lump–kink solutions, lump‐type solutions, high‐order lump solutions, and their interaction solutions 12–27 …”

Rational, semi-rational solution and self-consistent sources extension of the variable-coefficient extended modified Kadomtsev-Petviashvili equation

“…Soliton equations such as the Benjamin-Ono equation [15][16][17], the mKdV equation [18] possess algebraic solitons which are given by the rational solutions. Rational solutions also describe some physically significant nonlinear waves such as the explode-decay waves [19], rough waves [20][21][22][23][24] and lump waves [25][26][27]. Rational solutions for the integrable equations were first discovered by Airault et al in late 1970ʼs [28].…”

“…Rational solutions for the integrable equations were first discovered by Airault et al in late 1970ʼs [28]. Since then the rational solutions for the integrable equations have been investigated widely and different approaches have been developed to construct the rational solutions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. In this paper, we derive the Nth-order rational solution of the variable-coefficient extended mKP equation by using the Hirota bilinear method and determinant technique.…”