Dynamical Properties Of Some Nonlinear Semi-discrete Integrable And Nonintegrable Systems

In this dissertation,we study the dynamical properties of several nonlinear semi-discrete nonintegrable equations,discuss the explicit solutions to some nonlocal inte-grable(semi-discrete)nonlinear systems as well as dynamic behavior of these solutions and give out meanwhile gauge-equivalent structures for those equations.The main contents are summarized as follows:In chapter 1,we outline the research background associated with this dissertation including some solving methods for integrable systems,the concept of gauge equiva-lence between two integrable systems,the research advance in the nonintegrable semi-discrete nonlinear Schrodinger equations(NLS).Besides,we state the main results and innovation points of this thesis.In chapter 2,by using the concept of the prescribed discrete curvature,we prove the nonintegrable semi-discrete focusing Hirota(defocusing Hirota)equation is gauge equivalent to a nonintegrable semi-discrete spin system,which is a generalization of the semi-discrete(modified)Heisenberg ferromagnet equation.we will also study the dynamical properties for the two nonintegrable semi-discrete Hirota equations with the help of the planar nonlinear discrete dynamical map approach.The exact spatial period solutions of the two nonintegrable semi-discrete Hirota equations are obtained through the construction of period orbits of the stationary discrete Hirota equations.By using the gauge equivalence,we obtain the exact solutions to the nonintegrable generalized semi-discrete(modified)HF equation.We perform the numerical simulations for the stationary discrete Hirota equations.We find that their dynamics are much richer than the ones of stationary discrete NLS equations.In chapter 3,we study a nonintegrable semi-discrete NLS equation with nonlinear hopping.By using the planar nonlinear dynamical map approach,we address the spa-tial properties for the nonintegrable discrete NLS equation.Through the constructions of exact period orbits of a planar nonlinear map which is a stationary version of the nonintegrable discrete NLS equation,we obtain the spatial period solutions for the nonintegrable discrete NLS equation.We also give the numerical simulations of the orbits of the planar nonlinear map and show how the parameters affect those orbit-s.The stability of an orbit is determined by its residue.By using discrete Fourier transformation and Neumann iteration method,we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable discrete NLS equation.In chapter 4,we show that,under the gauge transformations,the nonlocal fo-cusing NLS and the nonlocal defocusing NLS are,respectively,gauge equivalent to a Heisenberg-like equation and a modified Heisenberg-like equation,and their discrete versions are,respectively,gauge equivalent to a discrete Heisenberg-like equation and a discrete modified Heisenberg-like equation.Although the geometry related to the non-local NLS and its discrete version is not very clear,from the gauge equivalence,we can see that the properties between the nonlocal NLS and its discrete version and NLS and discrete NLS have significant difference.By constructing the Darboux transformation for discrete nonlocal NLS equations including the cases of focusing and defocusing,we derive their discrete soliton solutions.In chapter 5,the gauge equivalence between the modified Landau-Lifshitz equa-tion and the perturbed defocusing nonlinear Schrodinger equation((NLS-))is proved from the perspective of geometry of given curvature condition.We obtain the first-order approximate 1-soliton solutions to the modified Landau-Lifshitz equation based on the soliton perturbation theory of the NLS-equation under corresponding gauge transformation.