Qualitative Analysis Of Some Nonlinear Differential Equations

Nonlinear differential equations appear in many fields of mathematics, as well asother branches of science such as physics, mechanics, material science and biologicalsciences etc. In nonlinear sciences many scientists and mathematician attracted a lot ofinterest in nonlinear differential equations due to complexity and challenges in theirtheoretical study.There are various questions which can be explored for nonlinear differentialequations: the existence of particular types of solutions, such as equilibrium solutions,time-periodic solutions, travelling waves, self-similar solutions, exact solutions, thedynamic stability of these solutions; the long time asymptotic behavior of solutions;blow-up phenomena; extinction behavior of solutions; chaotic dynamics; completeintegrability and so on.This thesis is concerned with the study of, blow-up phenomena, extinction property,exact solutions and properties of new chaotic dynamical systems, rely on the followingproblemsI. Non-local reaction diffusion problem with time dependent coefficientII. Quasilinear parabolic equation with absorption and nonlinear boundary conditionIII. Nonlinear parabolic equation with a gradient termIV. Biswas-Milovic equationV.(2+1)-dimensional hyperbolic nonlinear Schr dinger equationVI. New three-dimensional chaotic systemChapter1includes the introduction of the main topics and outline of thesis.In Chapter2, we study the blow-up phenomena for problems (I) and (II). We derivethe conditions on the data of these problems sufficient to guarantee that blow-up willoccur, and obtain an upper bound fort*. Also we give the conditions for globalexistence of the solution.In Chapter3, we discuss the extinction property of solution of problem of type(III), and derive the precise decay estimates of solutions before the occurrence ofthe extinction.In Chapter4, we find the new exact solutions for problems of type (IV) and (V), bythe aid of Adomian decomposition method. The performance of this method is found tobe reliable and effective and can give more solutions, which may be important for the explanation of some new nonlinear complex physical phenomena.In Chapter5, the new three-dimensional chaotic system is proposed, which relieson a nonlinear exponential term and a nonlinear quadratic cross term necessary forfolding trajectories. Basic dynamical characteristics of the new system are analyzed.Compared with the Chen system, the equilibrium points of the new system does notcontain the origin, and has a greater positive Lyapunov index can produce morecomplex shaped chaotic attractor.