Exact Solutions Of Nonlinear Equations And Convexity And Smoothness In A Kind Of Space

This dissertation has mainly done the following three aspects research: First, with the aid of symbolic computation and Wu method, the exact solutions of some nonlinear differential-difference equations have been studied. The hyperbolic function rational expansion method and the rational formal expansion method for them are put forward. And elliptic function rational expansion method for nonlinear evolutional equations is developed. Next, in Hirota bilinear operator extended to the supersymmetrical situation, many important supersymmetrical bilinear identical equations have been produced. And we applied them to obtain Backlund transformation and solitary wave solutions. Finally, for the following studies and in order to study the nonlinear problems in a widespread space, the relationship between the convexity and the smoothness is studied in the locally convex space.Chapter 1 mainly introduces the origin and development of several subjects related to this dissertation (including soliton theory, mechanization, the convexity and the smoothness in the locally space et. al.), as well as the work and achievements of the domestic and foreign scholars which have been obtained in these aspects. Our main works are presented at last.Chapter 2 is devoted to AC=BD model and its applications in nonlinear evolutional equations. First, basic notations, basic theory of C-D pair and C-D integrable systems to construct C-D pair are given out. Then, the theory of AC=BD is applied to differential-difference equations and bilinear formal differential equations. These greatly enlarge AC = BD theory and increase the new richer contents.In Chapter 3, the travelling wave solutions, soliton solutions, periodic solutions of differential-difference equations are studied based on symbolic computation Maple. The hyperbolic function expansion method is extended and the hyperbolic function rational expansion method is brought forward. Moreover rational formal expansion method is put forward. These methods are applied to every kinds of Toda lattice equations, Hybrid lattice equation, Ablowitz-Ladik lattice equation and Volterra lattice equation and many explicit exact solutions are obtained.Based on the ideas of solving nonlinear evolution equations, algebraic method, algorithm re-ality, mechanization, Chapter 4 extends the Jacobi elliptic function rational expansion method. With the help of symbolic computation and Wu method, more new explicit exact solutions, including soliton solutions, two periodic solutions, periodic solutions, of an asymmetric Nizhnik-Novikov-Veselov equation, the Davey-Stewartson equation and a generalized Hirota-Satsuma coupled KdV equations are constructed.In Chapter 5, the definition and properties of Hirota bilinear operator are briefly introduced. It is extended to the supersymmetric equations and many new supersymmetric identical equations are given out. Under the physical meaning, the N=l supersymmetric Sawada-Kotera-Ramani is put forward. Using Hirota's bilinear operator, a Backhand transformation and super-soliton solutions are obtained.Chapter 6 introduces the concept of the S-simplest form of a seminorm family P and that of the P-reflexive locally convex space (X, Tp). The seminorm family P and every its S-simplest form not only generate the same locally convex separated topology on X but also have the same convexity and smoothness. Moreover, relationship between the P-reflexivity and the reflexivity are discussed, which shows that the P-reflexivity and the reflexivity are two equivalent concepts when X is a normed linear space. Under the condition of P-reflexivity, we establish that a dual pair (X, P) is uniformly smooth (uniformly convex) if and only if its strong dual pair (X1. P') is uniformly convex (uniformly smooth).