Non-local Symmetry And Conservation Laws With A Sharp Soliton Solution Integrable System

Nowadays there appeared more and more nonlinear phenomena in the physical and social sciences. The differential equations which described such phenomena have attracted great attention of related physicists and mathe-maticians.In this thesis, we are mainly concerned with the non-local symmetries and exact solutions of several partial differential equations, such as two-component ?-Camassa-Holm equations, Foursov equation, complex Camassa-Holm equa-tion, Degasperis-Procesi equation, Novikov equation. In addition, we consid-ered the 3×3 spectral problem of the shallow water wave. The infinite number of conservation laws of Degasperis-Procesi equation and Novikov equation were constructed. Finally we proved that Degasperis-Procesi equation and Novikov equation admitted no non-local symmetries depending on pseudo-potential functions.The detailed work in this thesis is summarized as following:Firstly, we introduced the research background of the equations men-tioned above and the fundamental theories and methods which are applied to solve the non-local symmetries and exact solutions of the above equations. Then we presented our main work in this thesis.Secondly, we discussed the non-local symmetries and conservation laws of two component ?-Camassa-Holm equation, which is the short wave limit of the two-component Camassa-Holm equation. At first from the geomet-ric integrability of this equation, we obtained the pseudo-potential functions, then constructed the conservation laws and derived an infinite number of conservation densities. Then based on its trivial Lie-symmetry and partic-ular structure, we dealt with its non-local symmetries. Finally we obtained the non-local symmetries depending on the potential functions of two com-ponent ?-Camassa-Holm equation. Secondly, we considered the ?-Camassa-Holm equation with low dispersive item ?ux. Based on its Lax pair and wave functions, the pseudo-potential function is constructed. From this pseudo-potential function, we derived the conservation laws and infinite conservation densities, and then applying the method of Lie symmetry obtained the non-local symmetries depending on the pseudo-potential function. Thirdly, the modified ?-Camassa-Holm equation is presented as the non-local form of the modified Camassa-Holm equation. From the geometric integrability of this equation, we got the pseudo-potential function, then constructed the conser-vation laws and proved that this equation admitted no non-local symmetries depending on pseudo-potential function. At last, the short pulse equation is the scale limit of the modified ?-Camassa-Holm equation with linear item ux. We derived the pseudo-potential function, then constructed an infinite number of conservation laws.Thirdly, based on the invariant subspace method and a generalized con-ditional symmetry method, we were concerned with the exact solutions of b-family system. This solution is generated from 3-dimension invariant sub-space and corresponded to a kind of a generalized conditional symmetry. The functions in which the coefficients are depending on the time satisfy the Em-den equation. By analyzing the structure of the solutions of Emden equation, we obtained the structure of the solutions of this equation, gave the global ex-istence and the conditions of blow-up and furthermore described the structure of the solutions completely. Then we discussed the peaked solitons of the dual Foursov equation. From its weak form, we proved that this equation admits the peakons. Finally the complex Camassa-Holm system is considered. This system is derived from the Mobus curve flow and admits peaked solitons.Fourthly, we considered the non-local symmetries and the infinite num-ber of conservation laws of Degasperis-Procesi equation and Novikov equation. These two equations are classical wave equations with the 3×3 spectral prob-lem. On the basis of the construction of the non-local symmetries and the infinite number of conservation laws of the 2×2 spectral problem, we extended it to the 3×3 spectral problem. At first, we obtained two potential func-tions and two pseudo-potential functions. Then by extending the potential functions to the exponent series and comparing the coefficients, we obtained the over-determined equations by applying Lie's method. From the analysis of these equations, we deduce that Degasperis-Procesi equation admitted no non-local symmetries depending on its pseudo-potential functions. Likewise, for Novikov equation, we calculated its potential functions and pseudo-potential functions. Then by same operations, we know that Novikov equation admit-ted no non-local symmetries depending on its pseudo-potential functions.Finally, some open problems were given.