Conservation law and Lie symmetry analysis of the (1+1) dimensional dispersive long-wave equation

In this paper, we mainly study the integrability of 1+1 dimensional dispersive long-wave equation. Firstly, the Lie symmetry analysis of the equation is carried out in the first part. And the optimal system of the equation is obtained according to the symmetry, and the invariant solution and the reduced form of the target equation are solved according to the results. Secondly, we use different methods to solve the conservation law of the target equation. To begin with, we give the adjoint determination equation and adjoint symmetry of the 1+1 dimensional dispersive long-wave equation, and use the adjoint symmetry as the equation multiplier to find several conservation laws. Then we get a Lie bracket by using the relationship between the symmetry of the equation and the adjoint symmetry. Next its strict self-adjoint property is verified, and its conservation laws are solved by Ibragimov's method. Finally, the conservation laws of the target equation are solved by Noether's theorem. Thirdly we calculate some exact solutions of the target equation by three different methods. In the end of the paper, the Hamiltonian structure of the target equation, the generalized pre-symplectic that maps symmetries into adjoint-symmetries and some of its soliton solutions are calculated. In conclusion, we use the direct construction of conservation law method, Ibragimov's method and so on to solve some new conservation laws of 1+1 dimensional dispersive long-wave equation, use the relationship between symmetry and adjoint symmetry to construct the corresponding Lie brackets, and obtain some linear soliton solutions according to the conservation law of the equation.