Nonlinear Waves, Geometrical Integrability And Group Classifications

In this dissertation, under the guidance of mathematical mechanization proposed by famousmathematician Wu Wentsun and by means of many types of constructive transformations aswell as symbolic computation, some topics in nonlinear waves, integrable systems and grouptheoretical analysis of differential equations are studied from the points of view of geometry andalgebra, including exact solutions, Darboux transformations, non-isospectral evolution equationswhich describe pseudo-sphere surfaces, symmetries group classifications, additional equivalencetransformations, classical Lie reduction and classifications of conservation laws.Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear partial differentialequations. Firstly, the basic theories of AC=BD model and C-D pair are introduced, and thenthey are extended to studying the form-preserving transformations of (1+1)-dimensional partialdifferential equations. Secondly, we choose some examples to illustrate them in Chapter 3. (ⅰ)Based on a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term,an extended first kind elliptic sub-equation method is proposed to obtain solutions of nonlineardifferential equations. Many interesting exact solutions of generalized Zakharov equations are ex-plored, including new bell and kink profile solitary wave solutions, bright and dark solitary wavesolutions, triangular periodic wave solutions and singular solutions; (ⅱ) a variable-coefficientprojective Riccati equation method is presented to obtain non-travelling wave solutions for the(2+1)-dimensional generalized Broer-Kaup system; (ⅲ) Three kinds of explicit N-fold Dar-boux transformation of four (1+1)-dimensional soliton systems are constructed. Then thesetransformations are used to derive explicit (2N-1) and (2N)-soliton solutions of these systemsand the (2+1)-dimensional Kadomtsev-Petviashvili equation as well as modified Kadomtsev-Petviashvili equation. The explicit formulas of both the Darboux transformations and solitonsolutions are expressed by Vandermonde-like determinants which are remarkable compactnessand transparency.Recent years there are increasing interests in non-isospectral evolution equation, i. e.,the corresponding spectral problem with a time-dependent spectral parameterη, in the the-ory of soliton and integrable system. In chapter 4, characterizations of evolution equationsu_t=F(x, t, u, u_x, ..., u_x^k) and u_xt=F(x, t, u, u_x, ..., u_x^k) which describe pseudo-spherical surfacesare given, under a priori assumption thatηis differential function of x, t, thus providing asystematic procedure to determine a non-isospectral linear problem for which the given non- isospectral evolution equation is the integrability condition. It also paves a way form the pointof view of geometry to solve one of the central problems of integrable systems: To determinea given nonlinear differential equation integrable or not in Lax sense, i. e., whether it can bewritten as an integrablity condition of a pair of linear problems. The aforementioned subjectsform the first part of this dissertation.Group classification of differential equations, especially complete group classification, isone of the classical and very tough problems in the field of group theoretical analysis of dif-ferential equation. In chapter 5, complete group classification of a class of variable coefficient(1+1)-dimensional nonlinear telegraph equations f(x)u_tt=(H(u)u_x)_x+K(u)u_x, is given, byusing a compatibility method and additional equivalence transformations. A number of newinteresting nonlinear invariant models which have non-trivial symmetry algebra are obtained. Itis shown that the symmetry algebra is at most five-dimensional. As an application, the groupclassification of nonlinear telegraph equations u_tt=(H(u)u_x)_x+K(u)u_x is also provided. Fur-thermore, the possible additional equivalence transformations between equations from the classunder consideration are investigated. Exact solutions of special forms of these equations arealso constructed via classical Lie method and generalized conditional transformations. Localconservation laws with characteristics of order 0 of the class under consideration are classifiedwith respect to the group of equivalence transformations.Chapter 6 deals with the group classification of general KdV-type nonlinear evolutionequations of the form u_t=F(t, x, u, u_x, u_xx)u_xxx+a(t, x, u, u_x, u_xx) invariant under at mostfour-dimensional Lie algebra, by using the classical infinitesimal Lie method, the technique ofequivalence transformations and the theory of classification of abstract low-dimensional Lie alge-bras. It is shown that there are three equations admitting three dimensional simple Lie algebras,what's more, all the inequivalent equations admitting simple Lie algebra are nothing but them.Furthermore, we prove that there exist two, five, twenty-nine and twenty-six inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvableLie algebras, respectively. Chapter 5 and 6 form the second part of this dissertation.