Symmetry Reductions And Constructive Methods Of Nonlinear Evolution Equations

In this dissertation, under the guidance of mathematics mechanization and AC=BD the-ory, and with the aid of symbolic computation, some researches on symmetry reductions of the nonlinear evolution equations in mathematical physics and soliton equations and Backlund transformations constructed from the splitting of the Lie algebra sl(2) are discussed.Chapter1is devoted to introducing the history and development of mathematics mech-anization, soliton theory and symmetry analysis of differential equations. Some works and achievements on these subjects involved in this dissertation arc presented at home and abroad. Finally, the main work of this dissertation is listed.In Chapter2, the basic theories of AC=BD proposed by Professor Zhang Hongqing is introduced, as well as its applications in symmetries of the nonlinear evolution equationsIn Chapter3. at first, based on Lou’s direct method, the symmetry transformations of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation are presented from its Lax pair. Moreover, we obtain the admitted infinitesimal generators of the CDGKS equation from the symmetry transformations, and reduce the CDGKS equation with the obtained symmetries. We extend the direct method to two (2+1)-dimensional differential-difference equations, and present some new soliton-like solutions and periodic-like solutions by means of the symmetry transformations. Also we discuss the relations between the new solutions and old solutions for Toda lattice. Second, we use the classical Lie group method to study the symmetries reduc tions of (3+1)-dimensional equations, and give some group-invariant solutions with arbitrary functions due to the auxiliary equation method. At last, we use the classical Lie group method to get the symmetry reductions of a nonisospectral (2+1)-dimensional breaking soliton equa-tion and its Lax pair by considering the spectral parameter as an additional field. and obtain new reduced equations with their nonisospectral or isospectral Lax pairs. Based on one of the reduced Lax pairs, we give an explicit solution of the breaking soliton equation by Darboux transformation.By means of the differential form method, we consider the Lie symmetries of some differential-difference equations in Chapter4. Since the set of differential forms as given are not unique, we find the Lie symmetries of the (2+1)-dimensional Toda-like lattice from two differ- ent sett of discrete differential forms. respectively. Moreover it is shown that. the determining equations for the two sets give the same symmetries. and the set of differential forms for the lower-dimensional space can make the computation for finding symmetries simpler than the other. Then we use the differential form method to seek Lie symmetries for the Lax pair of a (2+1)-dimcnsional Camassa-Holm (CH) system, and obtain some reduced (1+1)-dimensional equations with their new Lax pairs and the conservation laws for the CH system.In Chapter5, based on the method that generates a hierarchy of commuting flows from a splitting of the Lie algebra sl(2). a new nonisospectral (2+1)-dimensional soliton equation and its Lax pair are constructed. With the help of Backlund transformation for a splitting of sl(2). we get a soliton solution of the new equation. Then we extend the method to differential-difference cases, as well as the Backlund transformation, and obtain the famous Toda lattice and its explicit solution successfully.