The Symmetries,conservation Laws And Analytical Solutions Of Some Nonlinear Partial Differential Equations

In real life,many physical phenomena can be described by nonlinear partial differential equations(NLPDEs).And the analytical solutions(exact solutions and approximate solutions)of NLPDEs is a hot topic at present,for a specific and practical physical problem,solving the corresponding properties of NLPDEs or studying the properties of their solutions,will contribute to understand and explain the essence of the actual physical problems greatly,however,it is very difficult to solve the nonlinear equations.In addition,the study of the conservation laws is also a important issue in the field of mathematical physics,it is of great significance to discuss the stability of solutions of NLPDEs and the discussion of the integrability,linearization and numerical calculation and so on.Based on the symmetry method,the symbolic computation system Mathematica,Wu's method and symmetry computing package,we study the symmetries,conservation laws and analytical solutions of some NLPDEs.The first chapter,we introduce the development background,research status and related knowledge of symmetries theory and conservation laws briefly.The second chapter,we study the Kudryashov-Sinelshchikov equation with arbitrary parameters and the modified KdV equation with variable coefficients by using classical Lie symmetry method,give the symmetries,similarity reductions and some invariant solutions,finally,we present their conservation laws by using the adjoint equation method which is introduced by Ibragimov.The third chapter,we use the approximate symmetry method proposed by Baikov,Gazizov and Ibragimov to study the approximate Lie symmetries of the coupled GearGrimshaw system,construct the one dimensional optimal system and some new approximate invariant solutions,construct its approximate conservation laws by using the partial Lagrangian method proposed by Johnpliai,Kara and Mahomed.The fourth chapter,we construct the lump solutions of(3+1)-dimensional BoitiLeon-Manna-Pempinelli equation and a(3+1)-dimensional nonlinear evolution equa-tion(NLEE)based on the Hirota bilinear method which is proposed by Hirota.The fifth chapter,the calculated results can provide useful information for the future research.Therefore,at the end of the paper,we discuss and summarize the research contents,and look forward to the further research work.