Lie Group Analysis And Analytical Investigations Of Some Nonlinear Models In Plasmas And Heisenberg Ferromagnetic Spin Chains

In nature,there are a variety of wave phenomena.The wave patterns in the plasmas are related to the phenomena including the effects of solar wind on the Earth's magne-tosphere,solar flare and stellar evolution.Spin waves,the collective excitations of the electron spin systems in the ferromagnetic materials,are used in the telecommunication systems and radars.For better understanding the nonlinear waves in the plasmas and ferromagnetic spin chains,people find it necessary to establish proper nonlinear mod-els.This dissertation aims to study the Lie symmetry groups and analytical solutions of several nonlinear models arising in the plasmas and a Heisenberg ferromagnetic spin chain.The main content of this dissertation is presented as follows:In chapter 1,we introduce the nonlinear phenomena,as well as common nonlinear models in plasmas and Heisenberg ferromagnetic spin chains.We also present the re-search progress of symmetries and conservation laws for the nonlinear models.Besides,the mathematical methods and arrangement of in this dissertation are given.In chapter 2,we investigate a(3+1)-dimensional modified Zakharov-Kuznetsov(mZK)equation,which describes the nonlinear plasma-acoustic waves in a magne-tized electron-ion plasma.With the aid of the Hirota method,bilinear forms and one-,two-and three-soliton solutions are derived.The characteristics and interactions of the plasma-acoustic solitons are discussed graphically.We present the effects on the plasma-acoustic soliton's amplitude by the nonlinear coefficients which are related to the ratio of the positive-ion mass to negative-ion mass,ionic number densities,initial densities of the lower-and higher-temperature electrons and ratio of the lower temperature to the higher temperature for electrons,as well as by the dispe.rsion coefficient,which is related to the ratio of the positive ion mass to the negative-ion mass and ionic number densi-ties.Moreover,using the Lie symmetry group theory,we derive the Lie point symmetry generators and corresponding symmetry reductions,through which certain analytic so-lutions are obtained via the(G'/G)expansion method.We demonstrate that such an equation is strictly self-adjoint based on which the conservation laws are derived.In chapter 3,we focus our attention on a(2+1)-dimensional mZK equation which describes the ion acoustic drift waves in an electron-positron-ion magnetoplasma.Lie symmetry generators and groups are presented by virtue of the Lie symmetry method.Optimal system of the one-dimensional subalgebras is presented.Based on the optimal system,we construct the power-series,multi-soliton,breather-like and periodic-wave solutions.Two types of the elastic interactions,including the overtaking and head-on interactions between(among)two(three)ion acoustic drift solitary waves are discussed.We find that the amplitudes of the ion acoustic drift solitary waves and periodic waves are positively related to the electron Debye length and negatively related to |?i| with pi as the ion Larmor radius.Besides,we find that the mZK equation is not only strictly self-adjoint but also nonlinearly self-adjoint.Based on the nonlinear self-adjointness of the mZK equation,conservation laws are obtained.In chapter 4,we work on the(3+1)-dimensional Zakharov-Kuznetsov-Burgers equa-tion for the ion-acoustic waves in a collisionless magnetized electron-positron-ion plasma.We derive the Lie point symmetry generators and Lie symmetry groups.Using the sym-metry generators,we obtain the reduction equations,through one of which we derive some power-series solutions and travelling-wave solutions including the shock solutions via the power-series and polynomial expansion methods.Ion-acoustic shock waves are pictured out.Effects of the normalized ion gyrofrequency,the normalized kinematic viscosity,the real parameter measuring the deviation from the Maxwellian equilibrium,the ratio of the ion temperature to electron temperature and the ratio of the electron temperature to positron temperature on the amplitude of the ion-acoustic shock wave are found.We present the conditions of the nonlinear self-adjointness.Based on the nonlinear self-adjointness,we construct the conservation laws of such an equation.In chapter 5,a(2+1)-dimensional nonlinear Schrodinger equation is investigated,which describes the nonlinear spin dynamics for a Heisenberg ferromagnetic spin chain.Lie point symmetry generators and Lie symmetry groups of that equation are derived.Lie symmetry groups are related to the time,space,scale,rotation transformations,and Galilean boosts of that equation.Based on the Lie symmetry generators,the re-duced systems of such an equation are obtained.Through one of the reduced systems,group-invariant solutions including the soli ton solutions are constructed.The n-th-order breather solutions are derived with n as a positive integer via the Darboux transforma-tion.We convert,under certain conversion conditions,the spin breathers to the lumps,rogue waves and two types of periodic waves which are named as the periodic-I and periodic-II waves in this dissertation.Via the second-order breather solutions,inter-actions of the two spin breathers,of the two periodic waves and of a spin breather and a periodic-I wave are graphically discussed.Through the theoretical and graph-ical analyses,it is found that the lumps and rogue waves are the long-wave limits of the spin breathers and periodic waves,respectively.Besides,the semirational solutions for describing the lumps and rogue waves on the periodic backgrounds are given.We discuss the effects of the coefficients of the bilinear exchange and neighboring interac-tions for the spins at the lattice sites,uniaxial crystal field anisotropy parameter and lattice parameter on the nonlinear spin excitations such as the magnetic solitons and spin breathers.In chapter 6,the main contents and innovations of this dissertation are given.Our further research works are presented.