Invariant Groups And Symmetries Of Nonlinear Equation With Self-consistent Sources

Nonlinear evolution equations can describe natural phenomena of fluid mechanics,plasma theory and nonlinear optical.Soliton theory is one of the important branches of Nonlinear Science,soliton equation with self-consistent source is a integrable coupling extension of the soliton equation.Based on the KdV equation with a self-consistent source and the nonlinear Schrodinger equation with a self-consistent source,the solutions of the source equation are studied and the propagation characteristics of solitary waves in a nonlinear system are investigated theoretically,The equations with self-consistent sources reflect the interaction between different solitary waves.In explaining the basic rule of natural phenomenon,the equations with self-consistent source terms are more abundant than the normal nonlinear equations.Firstly,the infinitesimal transformation and the corresponding vector space and the optimal systems are derived by Lie symmetry method.Further,the soliton solutions of the equations are obtained.Secondly,the exact solutions of the equations are constructed via the power series method.Specially,the interactions between soliton solutions are studied by adjusting the spectral parameters.Finally,we use the multiplier method to obtain the conservation laws of nonlinear equations with the self-consistent source.