Study On Weak Solutions For A Class Of Nonlinear Integrable Systems

In this paper,we consider weak solutions for the Novikov equation,the integrable modified Camassa-Holm equation and the three-component integrable Camassa-Holm system.Although these equations are similar to the classical Camassa-Holm equation in construction,they have some unique features.We firstly found an appropriate space,then construct the unique solutions to the three systems by the tech-nique of flow,and give the weak continuity(defined in 1.2)on the initial datum.Next,using priori estimate,we get the propagation of regularity for the solution in R.Mean-while,through smoothing and approximation,we have the propagation of regularity of the first two equations in some bounded open set.In the way of iteration,we prove,in some bounded open set,the propagation of continuity for highest order of derivation.This paper is organized in five chapters.In the first chapter,we give the study background for the Novikov equation,the integrable modified Camassa-Holm equation and the three-component integrable Camassa-Holm system,including the origin,physical significance and study value.And then we introduce some necessary definitions,notations and inequalitiesIn the second chapter,we consider the Cauchy problem of the Novikov equation.The difficulties are the nonlinear cubic term of the equation and the lack of regularity of initial dada.So for the sack of reducing the regularity,we should fist rewrite the equa-tion into a convolution form.Then after choosing a suitable flow,the Eular coordinate can be transformed into the Lagrange coordinate,and an integral system equivalent to the convolution form can be founded.Then we will construct the weak solution to the original equation and get its weak well-posdness(defined in 1.2)by the integral system.In the end of this charpter,we give the priori estimate,and discuss the propagation of regularity and continuity of highest order of the derivation of solution.In the third chapter,we consider the Cauchy problem of the three-component integrable Camassa-Holm system.However,the nonlinear cubic term becomes the main obstacle,which guides us to deduce the integral equation through the flow generated by²-².As for the construction of existing space,we should increase its regularity in some extent.Firstly,we get a weak solution,and then after a series of estimates,we have the regularity of the solution,as well as the continuous dependence on time variables and initial datum.Finally in the second section,we investigate the propagation of regularity in a bounded open set.In the fourth chapter,we consider the Cauthy problem of the three-component integrable Camassa-Holm system.We first rewrite the equation into an one-component convolution form.Then we study the existence,uniqueness,and the weak continuity for the solution.In the fifth chapter,we summarize our works and provide some ideas of further reseach.