Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures

This article mainly uses two methods of solving the conservation laws of two partial differential equations and a system of equations. The first method is to construct the conservation law directly and the second method is to apply the Ibragimov method to solve the conservation laws of the target equation systems, which are constructed based on the symmetric rows of the target equation system. In this paper, we select two equations and an equation system, and we try to apply these two methods to the combined KdV-MKdV equation, the Klein-Gordon equation and the generalized coupled KdV equation, and simply verify them. The combined KdV-MKdV equation describes the wave propagation of bound particles, sound waves and thermal pulses. The Klein-Gordon equation describes the nonlinear sine-KG equation that simulates the motion of the Josephson junction, the rigid pendulum connected to the stretched wire, and the dislocations in the crystal. And the coupled KdV equation has also attracted a lot of research due to its importance in theoretical physics and many scientific applications. In the last part of the article, we try to briefly analyze the Hamiltonian structures and adjoint symmetries of the target equations, and calculate their linear soliton solutions.