KAM Theory For Quasi-Linear Seventh-Order KdV Equations

In this paper,we prove the existence of small-amplitude quasi-periodic solutions for the quasi-linear KdV equations.We conjugate the linearized operator at an approximate solution to an operator with constant coefficients plus a bounded remainder,and use Kuksin’s lemma to solve the homological equation.The following is the specific organization of this paper:Chapter 1 mainly introduces the background of our problems,researches the status of the topic,and proposes and solves the problem.Chapter 2 gives some basic definitions,inequalities,and lemmas preparing for the later reducibility.Chapter 3 mainly introduces the specific reducibility process of linear operators.We eliminate the space and time dependence of the highest order derivative of linear operator by using change of variables.Then we get the linear operator whose highest derivative is the constant coefficient.Chapter 4 is the KAM iteration,which finally conjugates the operator to an operator with constant coefficients plus a bounded remainder,and gives the relevant estimates of the solutions.Chapter 5 is measure estimation using the measure lemma to estimate measures of the sets.Chapter 6 is the appendix giving the proof of some lemmas in this paper.