Bifurcations And Exact Solution Of Transmission Line Equation

As a branch of modern mathematics,nonlinear dynamic has become progressively important in natural and social sciences.However,despite the development of science and technology,numerous partial differential equations remain unsolved.There exist countless methods to study partial differential equations,each with their advantages and limitations.Therefore,an effective research method is a meaningful and valuable tool.In this paper,the bifurcation and exact solutions of a fifth-order KdV equation(the transmission line equation)are studied by using the basic theory and method of dynamical systems.Firstly,a traveling wave transformation is introduced to convert the partial differential equation studied into an ordinary differential equation,and subsequently obtain the corresponding Hamiltonian system.Secondly,the classification of parameters is discussed,and the eigenvalues of the Jacobian Matrix of the system under different classification conditions are analyzed.Thirdly,different branching parameters are determined according to different classification conditions,and a small perturbation is added to the branching parameters.The perturbed Hamiltonian system is converted to its normal form,then the center manifold theorem is used to reduce the dimension of the high-dimensional standard form,and the equilibrium point of the low-dimensional system is qualitatively analyzed.Phase portraits are drawn,the expressions of the obtained orbits are computed,and the corresponding wave solutions are displayed.Finally,this paper mainly solves the Hamiltonian system normal form and reduces the dimension of the high dimensional normal form under certain parameter classification.In some cases,the explicit expressions of the different orbits and wave solutions representations are obtained.In others,only the corresponding implicit expressions can be achieved.