Abundant multilayer network model solutions and bright-dark solitons for a (3 + 1)-dimensional p-gBLMP equation

Physical informed memory networks for solving PDEs: implementation and applications

“…In addition, most physical information deep learning methods construct numerical solutions of PDEs. In the [46], a neural network model based on generalized bilinear differential operators is proposed to solve PDEs [46]. The method obtains a new exact network model solutions of PDEs by setting the network neurons as different functions.…”

Research on sensitivity analysis and traveling wave solutions of the (4 + 1)-dimensional nonlinear Fokas equation via three different techniques

“…Reduction to ordinary differential equations is usually based on Lie symmetries and wave transformations. In literature recent years, lots of methods are given for solving NPDEs for example the tanh-coth strategy [1], the auxiliary equation technique [2,3], modified simple equation technique [4], Bernoulli function methodology [5], the new extended direct algebraic technique [6], the sine-Gordon expansion technique [7,8], Hirota bilinear technique [9], the simplest extended equation technique [10,11], the F-expansion technique [12], He's semi-inverse technique [13], the sub-ODE technique [14], the (G'/G) -expansion technique [15], the generalized Kudryashov technique [16], and many more. The common point of all the methods mentioned here is to convert the PDEs to the ordinary differential equations (ODEs) with the help of wave transformations.…”

Extended Calogero-Bogoyavlenskii-Schiff equation and its dynamical behaviors

“…Furthermore, Manakov et al discovered in [9] that the interactions of lump waves do not result in a pattern of phase changes. Regarding that, many powerful methods for finding the lump solutions of NPDEs have been developed over the past decades, including the long-wave limit approach [7,10], the nonlinear superposition formulae [11], the inverse scattering transformation [12,13], the invariance and Lie symmetry analysis [14,15], the Bäklund transformation [16,17], the bilinear neural network method [18][19][20][21][22][23][24], the Darboux transformation [25,26] and the Hirota bilinear method [27][28][29][30][31], Symbolic computation method [32][33][34][35] and other different methods [36][37][38][39][40][41][42][43]. Among the approaches stated above, taking a 'long wave' limit of the corresponding N-soliton solutions plays an important role in the investigation of M-lump solutions for nonlinear partial differential equations.…”