Study On Exact Solutions And Stability For Three Partial Differential Equation

Nonlinear partial differential equations have simulated many important phenomena in the fields of nonlinear optics,mechanics,chemistry and biology.In recent years,it has been found that fractional partial differential equations and high-dimensional partial differential equations can more accurately describe many practical problems in the field of science and engineering.Therefore,it is very important to find the exact solutions of fractional partial differential equations and high-dimensional partial differential equations and to analyze their properties.For the time-space fractional Modified Equal-Width equation in the definition of modified Riemann-Liouville derivative,the fractional complex transformation is used to transform it into an integer-order ordinary differential equation,and then the improved(G'/G)-expansion method is applied to obtain new exact solutions of the equation expressed by hyperbolic functions,trigonometric functions and rational functions with parameters.In order to study the properties of solutions intuitively,three-dimensional images of some traveling wave solutions with different parameter values are drawn by using numerical software Mathematica.For high-dimensional partial differential equations,new exact solutions of generalized(3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation and Integrable higher order Drinfeld-Sokolov-Satsuma-Hirota equation are obtained by using the auxiliary equation method.These solutions are expressed in the form of hyperbolic function,rational function,trigonometric function and exponential function.Based on the theory of modulation instability,the stability of solutions of these partial differential equations is studied.With the help of numerical software Mathematica,the three-dimensional image and one-dimensional image of some solutions under the corresponding parameters are drawn,and the dispersion relation graph is drawn.This is helpful to study the physical structure and stability of the solution,and provides a strong theoretical basis for practical application.The simplicity and effectiveness of the method are verified by an example.