Some Methods For Analytic Solutions Of Nonlinear Differential Equations And Integrable Systems

Based on the idea of computer mathematical mechanization and the unified theoretical of the model AC=BD, and by virtue of existing theories and some symbolic computation soft-wares, in this dissertation, we mainly focus on some topics from the view points of algebra and geometry, including the model AC=BD and "Trigram" structures of the soliton theory, the (Binary) Darboux and Backlund transformations、differential transform and nonlocal analysis of nonlinear differential equations and Hamiltonian integrable hierarchy, the finite genus so-lutions and integrable systems of nonlinear differential equations、supersymmetric equations and ultra-discrete equations, etc.In Chapter1, we introduce the history and development of computer mathematics and computer algebra, soliton theory, the mechanical algorithm and integrable system of nonlin-ear differential equation、supersymmetric equation and ultra-discrete equation in summary at home and abroad. At last, we present an outline of this dissertation.In Chapter2, based on the model AC=BD, the integrable system and the pair of C-D, we do some work in algebraic-geometry solution and Sato theory. Tn algebraic-geometry so-lution:we systematically derive the Its-Matveev formula, super-Its-Matveev formula. In Sato theory:we present the relationship between Lax equation and Sato equation, Lax equation and Zakharov-Shabat equation, Lax equation and inverse scattering scheme, Sato equation and Hirota’s bilinear equation, respectively, which can be used to construct a unified model of solv-ing soliton equation by using Tau function. To construct a unified and fundamental structure of soliton equations, it is the first time to introduce our "Trigram theory" including "Trigram structures" and "Trigram identities", which present some exterior-and interior-decomposition Trigram identities, respectively, to reveal some integrable systems generated by Wronskian, Grammian, Pfaffian, Schur functions and characteristic polynomials of Young diagram, Fock representation of Clifford-and Heisenberg-algebra, etc, from which the Fock spaces of Clifford (Heisenberg) algebra is a (isomorphism) Trigram space. Finally, we present a new approach to construct the relationship between Tau function and Theta function, which indirectly present the relationship between Trigram structure and algebraic-geometry solution. In Chapter3, based on the theories of Lax spectral problem and painleve singular-manifold method, we present three new kinds of N-fold Darboux transformations, auto-Backlund and bi-nary Darboux transformations, the corresponding periodic wave solutions and grammian solu-tions for a kind of differential equations. Using the discrete Lax spectral problem and choosing the appropriate spectral Vn(m), we present a new kind of Hamiltonian Lattice hierarchy, and fur-ther investigate some classical Lattice reductions, integrable in involutory Lax’s sense of the multi-Hamiltonian structure, discrete Darboux transformation and its analytic solutions. Based on the framework of Sato’s theory, a mKP equation with self-consistent sources (mKPFSC-Ss) and its Lax pairs are structured by virtue of the constrained mKP equation. Using the conjugate Lax pairs, we construct the forward, the backward and the N-fold binary Darboux transformation for the mKPESCSs which offers a non-auto-Backlund transformation between two mKPFSCSs with different degrees of sources. With the help of these transformations, some new classical solutions for the mKPFSCSs such as soliton solutions, rational solutions, breather type solutions and exponential solutions are found. Through research the theories of the dif-ferential transformation and the Pade approach technique method, we investigate the solutions with and without continuity at crest of Camassa-Holm equation. Compared to exact solutions, we also research the computational efficiency、high accuracy of the method.In Chapter4, by virtue of conservation law multiplier, we present the nonlocal analysis for a kind of differential equations, which include nonlocally related PDF systems, tree structure, nonlocal symmetry and conservation law etc. By virtue of the nonlocal symmetry, we further investigate the nonlocal linearization and propose a new generalized algorithm of solving in-variant solution. For special kind of PDEs, the relationship between nonlocal symmetry and nonclassical method on solving solutions are presented.We investigate the nonlocal analysis of famous nonlinear Koinpaneets equation(NLK) joint work with Professor George. W. Bluman etc. Using the result obtained in nonlocal analysis, we obtain a wider classes of previously unknown solutions of the NLK equation beyond those ob-tained by Professor Ibragimov solution. These new solutions do not arise as invariant solutions of the NLK equation with respect to its local symmetries, which are breaking the status of ex-isting only one class of trivial local solution for NLK equation since1956. In particular, for live classes of initial conditions, each involving two parameters, previously unknown explicit time-dependent solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. The two classes exhibit blow up behavior in finite time, and the other three classes exhibit quiescent behavior. As a consequence, it is shown that the corresponding nonlrivial stationary solutions are unstable. The nontrivial stationary solutions are also beyond those obtained by Professor Dubinov.In Chapter5, based on the theories of superspace, and by means of the properties be-tween Hirota bilinear and Riemann theta function, we present some Riemann theta function periodic wave solutions with the finite genus (?) and the analysis of limiting characteristics for a kind of nonlinear differential equations and supersymmetry equations, respectively. Us-ing these methods, we investigate Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK) equation,(2+1)-dimensional breaking soliton(DBS) equation and supersymmetric Korteweg-de Vries-Burgers(sKdV-Burgers) equation, etc. By virtue of the rational identities of theta functions, we present some N-theta function periodic wave solutions of a kind of discrete soliton equations. This method can be extended to the ultra-discrete space of the discrete equations and the theta functions, based on which, we can further obtain Ud-Riemann theta function periodic wave solutions with the same genus (?) for the corresponding udtra-discrete equations. As its applica-tion, we investigate a discrete modified Korteweg-de Vires(mKdV) equation and a generalized Toda lattice equation, etc.In Chapter6, by virtue of multi-dimensional Bell polynomials and super Bell polynomials, we present the integrability analysis of a kind of nonlinear differential equations and supersym-metry equations, respectively. While we also present the judgment conditions of integrability to these equation(s) to make be a kind of integrable system(s). Using these methods, we inves-tigate a kind of generalized variable-coefficient Kadomtsev-Petviashvili equation, fifth-order Korteweg-de Vries equation and sKdV-Burgers equation, etc, and further obtain some new re-sults of integrability. Moreover, using the theory of "max-plus" algebra and the compatibility condition of Lax pair system, we propose the ultra-discrete equation and its Lax integrability theorem and solvability theorem. Through researching the ultra-discrete process of Riemann theta function with finite genus (?) we further present the same genus of Ud-Riemann theta function solutions of a kind of ud-discrete equations. Finally, the generalized theories of Ultra-discretization and its integrability are applied to discrete Lattice Krichever-Novikov equation, discrete mKdV equation and discrete Painleve equation, etc.