Application of Bilinear Neural Network Method to Sawada-Kotera Model

Abstract

In this work, we use the bilinear neural network method (BNNM) for solving the (2+1)- dimensional bidirectional Sawada-Kotera (bSK) equation. This model is the extension of the famous Korteweg-de Vries (KdV) equation which represents the shallow water waves on surfaces. Firstly, we use generalized Hirota bilinear approach (gHBA) and then by us- ing BNNM, we obtain exact analytical solution. BNNM is an important step toward the integration of neural networks with classic methodologies for nonlinear partial differential equations (NLPDEs), presenting both the solution in exact form and offering powerful vi- sualization of complex phenomena. This may be a method applied to a wide range of fields such as plasma, optics, acoustics, fluid dynamics, and so on. BNNM models the neural network with a tensor formula and derives the exact solutions by symbolic computation. This neural network consists of layers, with their contribution in finding the solutions using activation functions and weight matrices. This method generates new test functions and in this thesis, the test functions consist of ”3-2-1” and ”3-3-1” neural network models. Here ”3-2-1” means 3 neurons in input layer, 3 neurons in first hidden layer and a single neuron in second hidden layer. Lump, lump with one kink, rogue wave, and breather lump soliton and periodic solution for the governing model will be obtained by BNNM with the men- tioned layer. Sev