| The symbolic computation is an interdisciplinary subject involv-ing mathematics, computer science and artificial intelligence. With the development of the computer hardware and software, it gradu-ally becomes possible to algorithmically deal with some mathematical deduction and algebraic computation on a computer. The solitons, which are a kind of spatially-localized wave solutions of nonlinear evolution equations (NLEEs) and admit the stable structures and elastically-interacting properties, are formed on the balance between the effects of dispersion and nonlinearity. In soliton theory, the sym-bolic computation has been mainly used to solve a large amount of al-gebraic calculations which are encountered in studying the integrable properties and analytic solutions of NLEEs. The task of studying soliton theory via symbolic computation is to develop constructive al-gebraic algorithms and integrate them into the symbolic computation packages. With the aid of symbolic computation, this dissertation an-alytically investigates some vector, coupled, variable-coefficient and higher-dimensional NLEEs, and propose some algebraic algorithms which can be performed on a symbolic computation system. Con-cretely speaking, the work of this dissertation includes the following six aspects:(1) Symbolic-computation-based study on the multi-component Wronskian representation and asymptotic analysis of the vector N-soliton solution. By the Darboux transformation (DT) and Cramer's rule, the N-soliton solution to the vector focusing nonlinear Schrod-inger (NLS) equation arising from some fields such as nonlinear fiber optics is found to be expressible as the multi-component Wronskian. The straightforward verification of such solution is finished by devel-oping some new multi-component Wronskian identities. In order to study the collision properties of vector solitons in a direct way, an al-gebraic procedure is also presented, which allows to derive the asymp-totic expressions for any given vector N-soliton solution as t t?(?)?. This procedure is used to obtain the asymptotic expressions of the vector two- and three-soliton solutions, based on which some fea-tures about the vector-soliton collisions are analyzed, including the parametric conditions for the amplitude preservation of all soliton components after collision, explicit phase-shift formulae induced by the vector-soliton collision and generalized linear fractional transfor-mations describing the state change before and after collision for each soliton component.(2) Symbolic-computation-based study on the coupled derivative NLS system and Alfven solitons with dual polarization in cosmic plasmas. The integrability in the sense of "admitting the Lax pair" for the coupled derivative NLS system is indicated by the construc-tion of two gauge-equivalent Lax pairs. With symbolic computation, the coupled derivative NLS system is bilinearized and the analytic one- and two-soliton solutions are constructed by the Hirota method. Based on the asymptotic analysis of the two-soliton solution, the col-lision dynamics of Alfven solitons with dual polarization is found to be characterized by three features:(a) The soliton velocity and width are the same before and after collision; (b) The amplitude-changing collisions occur in two polarized components of Alfven solitons under certain parametric conditions, along with the energy redistribution between two components; (c) The total energy of left- and right-polarized components of each colliding Alfven soliton is conserved during the collision.(3) Symbolic-computation-based study on the integrable condi-tions of variable-coefficient NLEEs and inhomogeneous soliton phe-nomena in optical fibers and plasmas. The Painleve test is used to determine the integrable conditions for a variable-coefficient N-coupled NLS system and a variable-coefficient derivative NLS equa-tion. For the former, the inhomogeneous one- and two-soliton so-lutions are obtained by the DT, and the discussions are focused on some envelope soliton excitations in the density-inhomogeneous plas-mas and the propagation and collision properties of soliton pulses in the inhomogeneous optical transmission system. For the latter, the inhomogeneous one- and two- Alfven-soliton solutions are also derived by the Hirota method, and the analysis is made for the influence of inhomogeneities on the propagation and collisions of Alfven solitons, as well as the energy radiation effects of collapsed Alfven solitons and their potential applications in cosmic plasmas.(4) Lax-pair-nonlinearization-based study on integrable decom-positions of the modified Kadomtsev-Petviashvili (KP) and (2+1)-dimensional Gardner ((2+1)-DG) equations. With some constraints between the potentials and eigenfunctions in a single Lax pair or two symmetric Lax pairs, the modified KP equation is respectively decomposed into the first two members of the multi-component Chen-Lee-Liu (CLL) and Kaup-Newell (KN) hierarchies, and the (2+1)-DG equation is respectively decomposed into the first two members of the generalized Burgers and multi-component mixed derivative NLS hierarchies. Then, the Lax representations and DTs for the multi-component CLL, KN and mixed derivative NLS hierarchies are re-spectively constructed, and the generalized Burgers hierarchy is lin-earized by the Hopf-Cole transformation. Moreover, two algebraic al-gorithms based on the DTs and binary nonlinearization of Lax pairs are presented, which can be performed on a symbolic computation system to generate a series of analytic solutions of the modified KP and (2+1)-DG equations in a recursive manner.(5) Symbolic computation and the extension of Clarkson-Kruskal (CK) direct method:(a) The similarity transformations from the (2+1)- and (3+1)-dimensional Burgers equations which have both arisen in fluid dynamics and astrophysics to the Burgers equation are directly constructed, so that all the solutions of the Burgers equation can be mapped into those of the (2+1)- and (3+1)-dimensional Burg-ers equations, respectively. Via those transformations, some families of higher-dimensional N-shock-wave-like solutions containing several arbitrary functions are particularly obtained, which can describe the non-traveling effects and coalescence phenomena of two- and three-dimensional shock waves. (b) Based on the idea of CK direct method (but the number of independent variables keeps the same), three fam-ilies of transformations from a generalized variable-coefficient NLS equation arising in inhomogeneous plasmas, nonlinear fibers, arteries and Bose-Einstein condensates to the standard one are obtained, to-gether with the relevant constraint conditions on the coefficient func-tions. The Lax pairs are also derived under the corresponding con-ditions. (c) By combination of the CK direct method and Hopf-Cole transformation, a forced Burgers equation with variable coefficients arising from fluid dynamics is linearized into the constant-coefficient heat equation with the satisfaction of certain constraint conditions, so that a family of N-shock-wave-like solutions are derived, which can describe the inhomogeneous and forcing effects and coalescence phenomena of shock waves.(6) Symbolic-computation-based study on the extension of the tanh-function method:An extension is proposed for the tanh-function method by introducing a general series expansion form which con-tains negative power terms and two arbitrary parameters. Based on the general expansion form and Ricatti equation, the extended tanh-function method can be used to construct more soliton solutions, trigonometric solutions and rational solutions for NLEEs. Such an algebraic algorithm is applied to the Whitham-Broer-Kaup model which describes the wave motion in shallow water, obtaining five new families of soliton solutions which include two types of double-peak soliton solutions with unequal amplitudes. |
PDF Full Text Request