Dynamicalanalysis Of Solutions Near The Soliton On Camassa-Holm Equation

In this paper,solutions of the Cauchy problem on Camassa-Holm equation near the soliton are decomposed by the pseudo-conformal transformation under the equivalent norm as follows:λ^1/2（t）u（t,λ（t）y+x（t））=Q（y）+ε（t,y）,and the estimation formula with respect toeis obtained:|ε（t,y）|≤Ca₃Te^-θ|y|+|λ^1/2（t）ε₀.We prove that the solution of the Cauchy problem and the soliton Q are sufficiently close for t∈[0,T),and the approximation degree of the solution and Q is the same as that of initial data and Q,besides the energy distribution ofeis consistent with the distribution of the soliton Q in H².Furthermore,the dynamical behaviors of solutions near the soliton are analyzed,and interesting results are obtained by choosing the appropriate quantity.The structure of this paper is organized as follows:In the first section,the research background and current situation of this paper as well as the main results are introduced.In the second section,some basic concepts involved,several important inequali-ties and related theorems in this paper are given.In the third section,solutions of the Camassa-Holm equation near the soliton by taking advantage of the exponential decay of solitary wave are studied.In the fourth section,the dynamic behavior of solutions near the soliton on Camassa-Holm equation is described by choosing the appropriate quantity.In the fifth section,the prospect of the future research by summarizing the full text is given.