Inverse Scattering Method Landau-Lifshitz Equation

In this dissertation, we consider the inverse scattering method of Landau-Lifshitz equation. There are lots of works concerned the integrable Landau-Lifshitz equation in the70s of the last century. Domestic scholars research the relevant issues of this model also in the80s and90s of the last century. Although the classical integrable model is idealization, it provides research idea of the other non-integrable models. Therefore, finding some new phenomenon and new problems in the integrable models can be used to define the research orientation for the non-integrable ones. The main tasks in this dissertation are researching the classical model by inverse scattering method or Riemann-Hilbert approach. On account of the recent developments of inverse scattering method, we can obtain some new results by this approach. Firstly, we can obtain the well-posedness in the frame of inverse scattering. Secondly, we take advantage of the generalized Darboux transformation to tackle with the discrete scattering data and exact solution. We combine generalized Darboux transformation with inverse scattering method for the first time. By this cohesion, we ingeniously deal with the high order pole scattering coefficient. Moreover, the general soliton formula and the complete classification of soliton solution can be obtained. Finally, we can analysis the long time asymptotics of this model by Deift-Zhou method. These results can not obtained by classical PDE analysis. Therefore, the inverse scattering method, as a method to solve the Cauchy problems of integrable partial differential equation, provides the important ways to research it. Recently, there is a hot topic-rogue wave in the nonlinear science. An important physical mechanism to explain the rogue wave is the modulational instability. Owning to the ill-posednees of the model, we can not use the theory of analysis of PDEs in this instance. The inverse scattering method provides new ways to tackle with it.In chapter1, we briefly introduce the physical background and historical results of the model. Besides, we present the historical background, some new results and important progress with respect to the inverse scattering method. The main innovative points are also presented in this part.In chapter2, we concern on the inverse scattering method of the classical integrable Landau-Lifshitz equation. Firstly, we obtain the well-posedness in a weighted Soblev space by combining the gauge transformation and some known results. Secondly, we present the completely classification of soliton solution by combining the generalized Darboux transformation and inverse scattering method. Finally, we analyse the long time asymptotics of Landau-Lifshitz equation by Deift-Zhou nonlinear steepest descent method.In chapter3, we concern on the spherical symmetry Landau-Lifshitz (ssL-L) equa-tion. The integrability of this model is given in by Lakshamnan group. There are some results about exact solution. But there is no result about the Cauchy problem about this model, because it is the non-isospectral half line problem. We need extend the spectral problem on the line properly. Besides, we analyse the dynamics of solution of ssL-L equation. As a by-product, we show the dynamics of the generalized NLS equation.In chapter4, we consider the inverse scattering problem of Landau-Lifshitz equa-tion on the spin wave background. We give the gauge transformation and inverse scattering analysis. Firstly, we take advantage of gauge transformation to convert this problem into focusing NLS equation on the plane wave background. Besides, we give the Galilean transformation to Landau-Lifshitz equation and conversation laws. Finally, we utilize the generalized Darboux transformation to give the generalized soli-ton formula. Moreover, the breather solutions and rogue wave solutions are presented explicitly. The dynamics of solution is obtained by plotting picture.