Exact Solution And Simulation Of A Class Of Nonlinear Evolution Equations

The nonlinear evolution equations and their solutions are important mathematical models to describe the nonlinear phenomena in many fields such as physics and engineering.In this paper,three equations with wide application background,namely the coupled nonlinear Schr?dinger equations with variable coefficients,the modified regularized Long-Wave equation and the stochastic KdV equation,are studied,and their exact or numerical solutions are obtained by using modified sineGordon equation expansion method and finite difference method,and numerical simulation is realized.In the first chapter,we mainly explain research background and significance,and then introduce the research status of three important equations as well as the arrangement and main work of this paper.In the second chapter,we introduce methods of this paper and some preliminary knowledge about finite difference method and Brownian motion.In the third chapter,based on the coupled nonlinear Schr?dinger equations with variable coefficients,we obtain a new solitary wave solution of the equation by using the modified sine-Gordon equation expansion method.On this basis,we simulate the shape change of the solitary wave solution when the coefficient of the group velocity dispersion term changes.In the fourth chapter,the Method of Lines is used to study modified regularized Long-Wave equation through three numerical experiments:the propagation of a single solitary wave,the interaction of two solitary waves,and the solitary wave generated by Maxwellian initial conditions and the error norm of L2,L?and the three invariants I1,I2 and I3 are calculated,and the numerical results obtained are compared with the exact solution and some results in the literature.In the fifth chapter,based on the stochastic KdV equation,first,the first order forward difference is used for the discretization of time direction,the center difference scheme is used for the discretization of space direction,and the discrete Brownian motion is used for the noise term.Then,the numerical solution generated by the described difference scheme is simulated and compared with the known exact solution.Further,the influence of noise with different amplitudes on the solitary wave solution is discussed.Finally,the summary and prospect of the research content of this paper.