|
Studies of various explicit exact solutions of nonlinear dispersive equations hadattracted much attention in connection with the important problems that arise in scientificapplications. Mathematically, these explicit exact solutions have been studied by usingvarious analytical methods, such as inverse scattering method, Darboux transformationmethod, Hirota bilinear method, Lie group method, bifurcation method of dynamic systems,sine-cosine method, tanh function method, Fan-expansion method, homogenous balancemethod and so on. Practically, there is no unified technique that can be employed to handleall types of nonlinear differential equations. The tanh method and the sine-cosine methodare ones of most direct and effective algebraic method for finding exact solutions ofnonlinear diffusion equations. In Chapter 3, the (N+1)-dimensional sine-cosine-Gordonequations are studied. The existence of kink (or anti-kink) traveling wave explicit exactsolutions and periodic traveling wave explicit exact solutions are proved, by using theextended tanh method. In Chapter 4, a class of nonlinear fourth order variant of ageneralized Camassa-Holm equation is studied by using the sine-cosine method extendedby A. M. Wazwaz. It is shown that the extended tanh method provides a powerfulmathematical tool for solving a great many nonlinear partial differential equations inmathematical physics. Finally, the summary of this thesis and the prospect of future studyare given.
|