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Nonlinear Waves And Intrgrable Systems

Posted on:2003-07-05Degree:DoctorType:Dissertation
Country:ChinaCandidate:Z Y YanFull Text:PDF
GTID:1100360065956261Subject:Computational Mathematics
Abstract/Summary:PDF Full Text Request
In this dissertation, with the aid of many types of constructive transformations and symbolic computation, some topics in nonlinear waves and integrable system are studied, including exact solutions, Painleve integrability, Backlund transformation, Darboux transformation, symmetry (similarity reduction), conditional symmetry, Lax integrable hierarchy, Liouville integrable N-Hamilton structure, constraint flow, involutive system, Lax representation, r-matrix, separation of variables and integrable couplings.Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear wave equations: Firstly, the basic theories of C-D pair and C-D integrable system are presented. Secondly, we choose some examples to illustrate them in Chapter 3. (1) Based on two types of Riccati equations, two kinds of new methods are proposed to obtain solutions of nonlinear differential equations. Twelve families of exact solutions of WBK equation are found by using one of two methods; (2) The homogeneous balance method is improved cind investigated to (2+l)-dimensional Broer-Kaup equation such that many families of new solutions are derived. (4) Based on the isospectral Lax pair of Riccati form for generalized KdV equation with the force term, new Darboux transformation and solitary-like wave solutions and rational solutions are obtained; (4) By constructing Darboux transformation and the superposition formula of generalized variable coefficients KdV equation with the force term, new single solitary-like wave solutions, double solitary-like wave solutions and rational solutions are found for (2+l)-dimensional generalized KP equation.Chapter 4 deals with Painleve integrability and Backlund transformation. Two types of (2+l)-dimensional generalized Burgers equations are shown to pass the Painleve test by using WTC method, and their Backlund transformations are obtained through Painleve truncating expansion. Cole-Hopf transformation is their special case.In Chapter 5, symmetries and exact solutions of nonlinear partial differential equations are studied. (1) The C-K direct method is extended to new Estevez-Mansfield-Clarkson (EMC) equation with fully dispersion term, E(m, n) equation, such that five symmetries are found. Moreover new solitary wave solutions of ?l,n) equation and compacton-like of E(m,m - 1) equation are obtained. In particular, new compacton solutions of E(3,2) equation and E(2,1) equation are derived; (2) The C-K direct method is extended to higher-dimensional case-(2+1)-dimensional generalized KdV equation so that eight families of new (1+1)-dimensionaJ symmetries are found which leads to P-I equation and P-II equation. Finally the cnoidal waves and dromion-like structures areconsidered; (3) The C-K direct method is extended to (2+l)-dimensional generalized Burgers equation such that eleven types of new (l + l)-dimensional symmetries and six kinds of new conditional symmetries are obtained.Chapter 6 concentrates on new Lax integrabie hierarchies of equations and Liou-ville integrabie N-Hamilton structures. The Tu's scheme is extended to the generalized Dirac spectral problem with an arbitrary function, the generalized Kaup-Newell spectral problem and the new 3x3 spectral problem with five potentials such that their Lax integrabie integrabie hierarchies of equations and Liouville integrabie Hamilton structures are obtained.Chapter 7 is devote to higher-order constraint flow, involutive system, Lax representation, r-matrix and the separation of variables. (1) r-rnatrix, new involutive system and involutive solutions of Bargman constraint flow of generalized Dirac hierarchy are found; (2) higher-order constraint conditions and integrabie constraint flows of Guo's hierarchy and their Lax representations and r-matrix axe given; (3) It is shown that first constraint flow of Dirac hierarchy is separability and its separation equation is presented.In chapter 8, we propose a new implicitly loop algebra. New Lax integrabie couplings of the famous TC hierarchy are obtained by using the new spectral pro...
Keywords/Search Tags:nonlinear wave equation, exact solution, C-D pair, Backlund trans-formation, Darboux transformation, the improved homogenous balance method, Riccati equation, Painleve integrabie, symmetry, integrability, Tu's scheme, Lax representation, r-matrix
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