Application Of Nonlocal Symmetry And Bilinear Method In Nonlinear System

By using symbolic computation, this dissertation studies the theory and application of symmetry, integrability, KP-hierarchy reduction, integrable discretization in nonlinear science. The main work is classified into four aspects:nonlocal symmetries and their applications are investigated in coupled nonlinear integrable systems; soliton solutions expressed by Pfaffian for a class of multicomponent soliton equations are derived by using Hirota bilinear method, and a program is developed to test Pfaffian-type solition solution; multi-dark soliton, mixed soliton and rational solutions of the multicomponent coupled Yajima-Oikawa(YO) system are studied via KP-hierarchy reduction method; integrable semi-discretization of the coupled YO system and its bright, dark soliton solution are constructed, and Pfaffian-type multisoliton solutions of the continuous and semi-discrete (complex) Sp(m)-invariant massive Thirring models(SMTM) are provided.Chapter 1 is an introduction to review the research background and the current situ-ation of symmetry theory, bilinear method and symbolic computation. The main work of this dissertation is also provided.In Chapter 2, nonlocal symmetries and their applications for the Hirota-Satsuma cou-pled Korteweg-de Vries(HS-cKdV) system and the modified Generalized Long Disper-sive Wave (MGLDW) system are investigated. Based on Lax pairs, nonlocal symmetries expressed by spectral functions are obtained. On the one hand, nonlocal symmetries are localized successfully and the finite symmetry transformation and similarity reductions are derived. Further, some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and Painleve waves are presented. On the other hand, the negative hierarchies and the integrable models both in lower and higher dimensions are constructed.In Chapter 3. multisoliton solutions to the multicomponent HS-cKdV and Ito equa-tions are investigated based on the Hirota bilinear method. This general Pfaffian-type soliton solutions are proved by Pfaffian techniques. Based on bilinear method and pfaf-fian technique, a Maple program Pfafftestl is developed to calculate general pfaffian and search for the bilinear forms of coupled modified KdV-type and coupled derivative modified KdV-type by using three-soliton condition.In chapter 4, on the basis of KP theory, multi-dark soliton, mixed multi-soliton and rational solutions of multicomponent coupled YO systems are presented by using bilinear method. Firstly, N-dark-dark soliton solutions in both the Gram type and Wronski type determinant forms are derived and proved. The dynamics analysis shows that dark-dark soliton collisions are elastic and there is no energy exchange among solitons in different components. Then, mixed (bright-dark) multi-soliton solution to one-dimensional mul-ticomponent coupled YO system is derived. In this case, the inelastic collision can only take place among short wave components when at least two short wave components have bright solitons. Finally, explicit rational solutions expressed by determinant for one-and two-dimensional multicomponent YO systems is constructed. The fundamental rational solution describes the localized lump and rogue wave, which have three different patterns: bright, intermediate and dark states. Nonfundamental rogue wave includes multirogue wave and higher-order rogue wave. In particular, two-dimensional intermediate and dark counterparts of rogue wave are found with the different parameter requirements.In chapter 5, an integrable semi-discrete analogue of the one-dimensional coupled YO system is proposed by using the bilinear technique. Based on the reductions of the Backlund transformations of the semi-discrete BKP hierarchy, both the bright and dark soliton solutions in terms of pfaffians are constructed. Multisoliton solutions for the in-tegrable (complex) SMTMs are provided in both continuous and semidiscrete cases. Al-though the integrable semi-discrete SMTM system can be obtained by virtue of discretiz-ing Lax pair, the same semi-discrete analogue is derived via the Hirota’s discretization approach.In chapter 6, the summary and discussion of this dissertation are given, and the outlook of future works is discussed.