The Unique Continuity Of The Cauchy Problem Of The Fifth-order Kadomtsev-Petviashvili II Equation

In this thesis,we consider the unique continuation property of the solu-tions to the Cauchy problem for a kind of nonlinear equations:the fifth-order Kadomtscv-Pctviashvili ?(KP?)equation.The unique continuation property is an important property of the solutions to the intcgrablc systems.The proof of unique continuation property of the nonlinear equations has been explored and improved constantly.It is generally known that the classical methods arc:applying Carleman estimates,Fourier transformation,Bessel potential operator and inverse scattering transformation.In this thesis,we mainly recall two meth-ods and apply the methods to prove the unique continuation property of the initial value problem associated with the fifth-order KP? equation.Firstly,we apply the Fourier transformation to prove that,if a sufficiently smooth solution to the initial value problem is supported compactly in a nontrivial time interval,then it vanishes identically.Secondly,we use the method of Carleman estimates to obtain that the sufficiently smooth solutions of the fifth-order KP? equation have compact support at two different instants of time are identically zero.The constant of this paper is arranged as follows:In the first chapter,we briefly introduce the concepts,significance of the unique continuation property and the progress of the methods for this property is also mentioned both at home and abroad.In the second chapter,the related symbols,the definition and the prelimi-nary theories required to solve the problem arc described concretely.In the third chapter,we prove the unique continuation property of the solutions to the fifth-order KP? equation by applying Fourier transformation.In the fourth chapter,by using Carleman estimates,we obtain the unique continuation property of the solutions of the fifth-order KP? equation.