Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Abstract

Recently, searching for exact solutions to nonlinear partial differential equations has gradually become a hot research topic. It is of great scientific research and application value to reveal the law of wave propagation, explain natural phenomena accurately and apply related technologies scientifically. In this paper, bilinear neural network method (BNNM) was employed to obtain some new exact analytical solutions to the (2 + 1)-dimensional Ito equation. Based on the Hirota form of Ito equation, we constructed activation functions f in various forms containing the exp(ξ), sin(ξ), cos(ξ), cosh(ξ) and squares of polynomial functions in multi-layer neurons such as [3-2-2] and [3-2-3] neural network models. The test function f in this work is a new expression. On the other hand, these solutions have not been studied yet. As a result, we obtained several new interaction solutions, such as periodic wave solution, breather solution and bright-dark solitons solution, etc. In addition, the corresponding 3D, density and contour plots of all the solution forms were drawn and their characters and dynamic behaviors were vividly demonstrated.

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

“…The construction of exact solutions for NLEEs has created a very interesting and dynamic field whose historical background goes back to the discovery of soliton solutions in 1834 [8]. Subsequently, many effective methods are presented to obtain exact solutions for NLEEs; the Hirota bilinear method [9][10][11][12][13][14][15][16][17][18][19][20], the Wronskian technique [9,21], the inverse scattering transformation [22][23][24][25][26], the Darboux transformation [27][28][29][30][31], the deep learning method [32][33][34][35], the binary Bell polynomials [36][37][38][39] and so on. The integrability of NLEEs can be regarded as the key benchmark to test their exact solvability.…”

Various nonlinear characteristics of breather/rogue waves and controllable interaction phenomena for a new KdV equation with variable coeffcients

“…Lv [29] obtained a series of nonlinear phenomena of breather/rogue waves and interaction phenomena on nonconstant backgrounds for two KP equations by using symmetry analysis and BNNM. Ma [30] employed BNNM to obtain some new exact analytical solutions to the (2+1)-dimensional Ito equation. Cao [31] studied the breather wave, lump type and interaction solutions for a high dimensional evolution model.…”