Exact Solutions And Oscillation Of Differential Nonlinear Equations

This dissertation has mainly done the following two aspects research: First, with the aid of symbolic computation and Wu method, the exact solutions of some nonlinear differential equations have been studied. Some methods for constructing the exact solutions of nonlinear evolution equations are presented and improved. These presented methods are realized on symbolic computation system Maple. Second, by use of Riccati transformation and integral average technique, oscillation of several nonlinear differential equations has been studied.Chapter 1 mainly introduces the origin and development of several subjects related to this dissertation (including soliton theory, mechanization, oscillation 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 solving of nonlinear equations. Basic notations, basic theory of C-D pair and C-D integrable systems and method of constructing C-D pair are given out. Some illlustrative examples are presented to show how to use the model.Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality and mechanization, in Chapter 3, we firstly present the new generalized tanh function method, and apply it to construct the exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation. Second, the generalized Q-deformed hyperbolic functions method is developed based on the Q-deformed hyperbolic functions. The efficiency of the method can be demonstrated on the Shallow long wave approximate equations. At last, we extend Jacobi elliptic function expansion method, with which more Jacobi elliptic function solutions of auxiliary equations are obtained. Then by use of the method, more new doubly -periodic solutions of a class of nonlinear differential equations such as the generalized Ito system, Zakharov-Kuznetsov equation, the coupled Drinfel'd- Sokolov -Wilson equations and the (2+1)-dimensional Davey-Stewartson equation are obtained. These solutions are degenerated to solition solutions under degenerated conditions.In chapter 4, using general Riccati transformation and integral average technique: we first investigate the oscillation of a class of nonlinear differential equations under quite general assumptions. The results that we obtained generalize and improve some known oscillation criteria. Second, interval oscillation of a class of nonlinear differential equations with forcing term is concerned. These obtained reults are based on the information only on a sequence of subintervals of [t₀,∞) rather than on the whole half-line and avoid the problem of old ones. So these results are more powerful. At last, some new oscillation criteria are established for the nonlinear damped functional differentional equations: which are different from known ones in the sense that they are based on a class of functionsΦ(t, s, r) and operator A^ρ(·; l, t) defined. Our results are sharper than some previous results and easy to apply.