Symbolic Computation On Some Problems Of Integrable System

Based on symbolic computation, the integrablility and symmetry of some significant nonlinear evolution equations in nonlinear mathematical physics are investigated by Bell polynomial, by Riccati-typed pseudopotential and by nonlocal symmetry, respectively. Meanwhile, a software package which be used for investigating the integrablility of KdV-typed equation is developed. Furthermore, based on the Tu scheme and source generation procedure, the self-consistent sources and infinite conservation laws of a kind of super integrable equation hierarchy are obtained. The main work is carried out as follows:In chapter1, an introduction is devoted to review the research background and the current situation related to the dissertation, which including integrable systems and sym-bolic computation. The main works of this dissertation are also illustrated.In chapter2, the Bell polynomial approach is extended to investigate the integrablil-ity of some (1+1)-dimensional,(2+1)-dimensional and variable coefficient nonlinear evo-lution equations. For the modified generalised Vakhnenko equation, its corresponding bilinear representation, bilinear Backlund transformation, Lax pair, N-soliton solutions and quasiperiodic solution are systematically obtained. Meanwhile, relevant properties of the solutions are illustrated graphically. Moreover, the integrablility of the generalized KdV-fKdV-typed equation and variable coefficient KdV-CBS-typed equation are inves-tigated by introduce an auxiliary variable and impose a subsidiary constraint condition. In particular, the Darboux covariant Lax pairs and infinite conservation laws of the two equations are obtained.In chapter3, based on the Bell polynomial approach, a systematic algorithm is pro-posed to obtain the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws of the KdV-typed equation. Based on this algorithm, a corresponding software package PDEBellll in Maple is developed. In particular, the software package PDEBellll is also suitable for the seeking of the bilinear form of the mKdV-typed equation.In chapter4, based on the Riccati-typed pseudopotential theory and nonlocal sym-metry theory, the integrablility and symmetry of some nonlinear evolution equations are investigated. The Lax pair, AKNS-typed Lax pair, auto-Backlund transformation and sin-gularity manifold equation of the variable coefficient fKdV equation are obtained by using the Riccati-typed pseudopotential theory. The nonlocal symmetries of the fifth order Lax equation and generalized KP equation are obtained by their Riccati-typed pseudopoten-tial and Lax pair. Meanwhile, their corresponding finite symmetry transformation and symmetry reduction are also investigated.In chapter5, The self-consistent sources and infinite conservation laws of a kind of super integrable equation hierarchy are obtained. For a kind of super integrable equa-tion hierarchy, its self-consistent sources are obtained by source generation procedure; Meanwhile, by introduce two new variable, its corresponding infinite conservation laws are obtained by change the spectral problem to a Riccati-typed equation.In chapter6, the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.