A Class Of Coupled Equations Of Soliton Solutions And Their Non-classical Symmetry

In the integrable system, shallow water wave equation, which is completely integrable, has very important research value, and attracts the mathematics and physics scholars’strong attention. In1834, Russell obtained the KdV equation when observing shallow water. It is proved that it belongs to the completely in-tegrable system and has smooth solitons, and moreover, wave form is almost the same in the interaction. In1993, Camassa-Holm equation was derived by Ca-massa and Holm. It is different from the KdV equation, which admits peakons, is a new sort of integrable system. Therefore, it has bi-Hamiltonian structures, Lax pairs and recursion operators. In1996, the modified Camassa-Holm equation was found as a new integrable system by Fuchssteiner and Olver and Rosenau, who used the approach of tri-Hamiltonian duality to the bi-Hamiltonian representa-tion of the modified KdV equation. In the year of2009, Novikov obtained a new integrable equation named after the Novikov equation, which was discovered in a nonlocal symmetry classification of partial differential equations with quadrat-ic or cubic nonlinearity. Regardless of the modified Camassa-Holm equation or Novikov equation, people have done a majority of research for the integrability, solitons and their stability, well-posedness and blow-up phenomenon, and has obtained great achievements.From the foregoing, completely integrable system is quite complicated, we should do further research on it. In the first several chapters by using properties of Green’s function and test function method, peakons and period peakons of a class of coupled equations is studied. Later, we study the equation from nonclassical symmetry.An outline of this thesis is as follows:Chapter1is the introduction, which mainly describes the relevant back- ground knowledge, development and application of the Camassa-Holm equation, modified Camassa-Holm equation and Novikov equation.Chapter2firstly introduces the methods and ideas about how to solve the solitary wave solutions of the coupled equations, and moreover, we verify its correctness. Thus, this method can be popularized.In Chapter3, we introduce the methods and ideas about how to solve the period peakons of the coupled equations, and verify its correctness. So, this method can be extensively used.In Chapter4, we introduce the concepts and properties relating to symme-try, and further study the nonclassical symmetry of the coupled equations.In Chapter5, concluding remarks and future research are discussed.