Applications Of Hirota And PINN Method On KS And Boussinesq Equations

The research on the exact solutions of nonlinear evolution equations(NLEEs)is an important issue in the nonlinear field.Based on the symbolic computing software Maple platform,the Hirota bilinear method is used to study a series of local waves and their interaction solutions of the 3+1 dimensional Kudryashov-Sinelshchikov(KS)equation.First,by taking specific parameters on the soliton solution expression of the KS equation,three different breather solutions are constructed.Then the lump solution and rogue wave solution of the KS equation are obtained through the long wave limit.The interaction solution between the lump solution and the linear soliton of the KS equation is derived.According to the asymptotic analysis,it is found that the interaction is a completely inelastic collision.Finally,an image of the exact solution is given,and the dynamic characteristics of the solution are discussed.In recent years,deep learning has been applied to a large number of fields,and significant results have been obtained.Recently deep learning has been applied to the numerical solution of nonlinear evolution equations,and an efficient and stable numerical simulation algorithm has been obtained,which has attracted the attention of a large number of researchers.In this paper,we use the physics informed neural networks(PINN)method to obtain the soliton solution of the 1+1 dimensional Boussinesq equation.First,we design our own feedforward neural network for the Boussinesq equation.The loss function is defined,and the appropriate activation function and optimization algorithm are selected.Then,we selected the initial value and boundary value data.After model training,we showed the final prediction solution obtained by the model.Finally,the prediction results of the one soliton solution and the two soliton solution prove the effectiveness of the PINN method,and the algorithm can accurately restore the dynamic behavior of the soliton solution of the Boussinesq equation.