A Study On Various Exact Solutions And Integrability Of Nonlinear Evolution Equation Based On Symbolic Computation

Nonlinear evolution equations are widely used to describe the nonlinear phenomena in shallow water waves,nonlinear optics,Bose-Einstein condensation,plasma and other fields.In recent years,finding exact solutions of nonlinear evolution equations has become a hot topic in soliton theory.With the development of soliton theory,many effective solutions have been proposed,such as the Hirota's bilinear method,the inverse scattering method,the Riemann-Hilbert method and so on.In the process of seeking exact solutions,there are often a lot of regular and repeated calculations.With the help of symbolic computation,the speed and accuracy of calculation can be improved,which is convenient for inspection and verification.Based on symbolic computation,this paper constructs and studies a representative(2+1)-dimensional generalized nonlinear evolution equation with the Hirota's bilinear method as the main research method.The specific research contents are as follows:(1)By using the Hirota's bilinear method,the analytic forms of the one-soliton,two-soliton and three-soliton solutions of the equation are explored.Using software to draw the image of one-,two-and three-soliton solutions and analyze their motion mechanism;(2)By virtue of the linear superposition principle of bilinear equation and the characteristics of resonance solution,judge that the equation does not have resonance solution;(3)With the help of the trial function method,for setting the forms of the solutions in the bilinear equation as mixing positive quadratic functions with an exponential function or with a hyperbolic cosine function,the analytical expressions of the interaction solutions of the lump-kink type and the lump-soliton type are obtained.By analyzing the expressions of the two solutions,the asymptotic properties of the interaction process are obtained,and speeds,limitations and extreme values of the solutions are studied.Based on the numerical simulation,the motion process of the two interaction solutions are studied and the action mechanism and dynamic characteristics of the solutions are analysed;(4)By means of the Hirota's bilinear method and bell polynomials,the bell polynomials type B?cklund transformation of the equation is obtained.From the transformation,the Lax pair is obtained,and the Lax integrability of the equation is determined.The series expansion of Lax pair is made,and the infinite conservation laws are derived.The innovations of this paper are as follows:(1)The analytical form,asymptotic properties and dynamic characteristics of soliton solutions,resonance solutions,lump-kink type interaction solutions and lump-soliton type interaction solutions are analyzed to show the diversity of the solutions;(2)By using the Hirota's bilinear method,the trial function method,the bell polynomial method and other methods to explore many exact solutions and integrable properties of the equation;(3)The obtained solutions have practical application value.For example,the lump-kink type interaction solutions can be used to explain the nonlinear phenomena in the fields of shallow water wave and nonlinear optics,and the lump-soliton type interaction solutions can be used to predict the emergence of optical strange waves,financial strange waves and so on.