Periodic Soliton Solutions For Nonlinear Integrable Systems

Nowadays,in scientific research,more and more physical models are not simple linear problems but more complex nonlinear problems.Different physical models have their own characteristics in the context of their respective disciplines,but they also have certain universality,which is reflected in the interpenetration and interconnection of various disciplines.With the development of modern science and technology,various nonlinear models have been established to better describe the phenomena in the research of typhoon,navigation,mechanics,optics,communication science,biological science,plasma and other fields.It is extremely complicated to solve these nonlinear models and scientists have not found a solution method applicable to all nonlinear partial differential models up to now.Through continuous study of some nonlinear models,scientists have found that there exists a class of nonlinear partial differential equations whose nonlinear terms and dispersion terms have a clever balance,and this kind of nonlinear partial differential equations have been noticed in many fields.Through the study of the physical properties of this kind of nonlinear partial differential equation,it is found that it has great application potential,so it is of great significance to study the solution method of nonlinear partial differential equation and its physical properties.This paper introduces the solving methods of various kinds of nonlinear partial differential equations,and then applies the bilinear method to solve two kinds of nonlinear partial differential equations.The following is a brief description of the main work:Firstly,by understanding the methods of solving nonlinear partial differential equations,a new periodic soliton solution for the integrable(1+1)dimensional KdV equation of order 7 is solved.The bilinear form of the equation is obtained by using bilinear method,and the Hopf-Coletransformation is made by using homogeneous equilibrium method.Let's set up the test function f,and the parameters in the test function are calculated by the symbol calculation software Maple.Combining with the actual situation,the changing process of the waveform is simulated and the waveform of this kind of model is obtained.Then,a new periodic soliton solution for the integrable(2+1)dimensional Boti-Leon-Manna-Pempinelli(BLMP)equation is solved by a similar method.The bilinear form of the equation is obtained by using the bilinear method,and the hopf-Coletransformation is made by using the homogeneous equilibrium method,which is different from the(1+1)dimensional 7th order KDV equation.Let's set up the test function f,and the parameters in the test function are calculated by the symbol calculation software Maple.The collision of the waveform is simulated by combining the actual situation,and the waveform characteristics are obtained.Through solving the nonlinear partial differential equations with different dimensions and orders,the application of bilinear method to solving the nonlinear partial differential equations is extended.The new interaction forms of(1+1)dimensional 7th order KDV equation and(2+1)dimensional Boti-Leon-Manna-Pempinelli(BLMP)equation are given for solving the two kinds of equations,which enrich the solutions of these two kinds of equations.