Research On Lie Symmetry Analysis And Bilinear Method For Integrable Systems

In this thesis,the Lie symmetry method and the bilinear method are employed to study the properties of some integrable systems with physical background.Based on the theory of the Lie symmetry analysis,three kinds of nonlinear integrable systems are analyzed systematically,and the specific content is as follows:The Heisenberg equation from the statistical physics is studied.Its Lie point symmetries are obtained.The method of constructing the optimal system of one-dimensional subalgebras by means of the commutation table is extended.Based on the optimal system,the similarity reductions and the group-invariant solutions are investigated.In view of the multipliers,three local conservation laws are obtained.The Heisenberg equation is nonlinearly self-adjoint.The conservation laws associated with symmetries of this equation are constructed by means of Ibragimov's method.The AKNS system from a member of the famous AKNS hierarchy is analyzed.Some results of this system are investigated systematically,such as the Lie point symmetries,the optimal system of subalgebras,the similarity reductions and the group-invariant solutions.Four local conservation laws are derived by using the direct method.It is proved that the AKNS system is quasi self-adjoint.Two non-trivial conservation laws are derived via utilizing the new conservation theorem.The Lie symmetry method is employed to investigate the Lie point symmetries and the one-parameter transformation groups of a(2+1)-dimensional Boiti-Leon-Pempinelli(BLP)system,which describes interactions of two waves with different dispersion relations.The more complex optimal system of subalgebras is further studied.The truncated Painleve analysis is used for deriving the Backlund transformation.The method of constructing the lump-type solutions of integrable systems by means of the Backlund transformation is presented.Meanwhile,the lump-type solutions of the BLP system are obtained.The dynamical behaviors of the lump-type solutions are discussed through the graphical analysis.The fusion-type N-solitary wave solutions are also constructed with the aid of the Backlund transformation.In addition,this system is solvable in terms of the CRE method.Based on the bilinear method,the Riemann theta function quasiperiodic wave solutions and the lump solutions of two nonlinear integrable systems are analyzed respectively,and the results are as follows:The Riemann-Backlund method is extended to the variable coefficient integrable systems.The soliton and the quasiperiodic wave solutions of a generalized variable coefficient(2+1)-dimensional KdV equation are investigated systematically.The relations between the quasiperiodic wave solutions and the soliton solutions are rigorously established by a limiting procedure.It is proved that the quasiperiodic wave solutions tend to the soliton solutions under a small amplitude limit.Furthermore,the propagation characteristics of the soliton solutions and the quasiperiodic wave solutions are summed up through the graphical analysis.The lump soliton solutions,the mixed lump stripe solutions and the periodic lump solutions of a(2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation from incompressible fluid are constructed by making use of the bilinear method.It is proved that the interaction between two solitary waves is nonelastic.The strip soliton finally drowns or swallows up the lump soliton with the evolution of time.The periodic lump wave can be regarded as a superposition of the single lump solitons.