![]() | Only 14 pages are availabe for public view |
Abstract This thesis deals with problems arising in the study of nonlinear partial di¤erential equations (PDEs). Many methods have employed for the analytic treatment of the non- linear partial di¤erential equations. This thesis contains six chapters. In the introductory chapter one, we survey the methods used to generate exact traveling wave solutions for the nonlinear partial di¤erential equations. In chapter two, we apply four di¤erent methods to nd the exact traveling wave solutions of some nonlinear PDEs. Firstly, we use the extended tanh-function method to construct the exact traveling wave solutions of the Biswas-Milovic equation with dual-power law nonlinearity. With the aid of computer algebraic system Maple, both constant and time- dependent coe¢ cients of the nonlinear Biswas-Milovic equation are discussed. Secondly, we use the modi ed extended tanh-function method and the Bäcklund transformation of the Riccati equation to construct the exact traveling wave solutions of the generalized KdV-mKdV equation with higher-order nonlinear terms. Finally, we use the Bäcklund transformation of the generalized Riccati equation to construct the exact traveling wave solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation. In chapter three, we apply the two variable (G0 G ; 1 G)-expansion method to nd the ex- act traveling wave solutions of some nonlinear PDEs, namely, the equation of nano-ionic currents along microtubules (MTs), the higher order nonlinear Klein-Gordon equations, the higher order nonlinear Pochhammer-Chree equations, the (2+1)-dimensional nonlin- ear cubicquintic GinzburgLandau equation (CQGLE), the (1+1)-dimensional resonant nonlinear Schrödingers equation with dual-power law nonlinearity, the (1+1)-dimensional Schrödinger-Boussinesq system (SB-system), the (2+1)-dimensional hyperbolic nonlinear Schrödinger (HNLS) equation, the higher order nonlinear Schrödinger equation which de- scribes the propagation of ultrashort femtosecond pulses in nonlinear optical bers and the higher order nonlinear Schrödinger equation which describes the propagation of femtosec- ond pulses in nonlinear optical bers. Exact traveling wave solutions of these nonlinear PDEs include kink, anti-kink soliton wave solutions, bell and anti-bell soliton wave solu- tions as well as periodic and rational wave solutions. In chapter four, we apply di¤erent methods to nd the exact traveling wave solutions of some nonlinear PDEs. Firstly, we nd the exact traveling wave solutions of the nonlinear PDE describing pulse narrowing nonlinear transmission lines by using the new Jacobi ellip- tic function expansion method. Secondly, we nd the exact traveling wave solutions of the nonlinear PDE governing wave propagation in nonlinear low-pass electrical transmission lines by using the new Jacobi elliptic function expansion method and a direct method. Finally, we apply the extended auxiliary equation method to nd the exact Jacobi el- liptic function solutions for a class of nonlinear Schrödinger-type equations namely, the nonlinear cubicquintic GinzburgLandau equation, the resonant nonlinear Schrödingers equation with dual-power law nonlinearity and the nonlinear generalized Zakharov system of equations. The exact solutions of these nonlinear PDEs include terms of the hyperbolic or the trigonometric functions when the modulus of the Jacobi elliptic function m ! 1 or m ! 0 respectively. In chapter ve, we apply the generalized projective Riccati equations method to nd the exact traveling wave solutions of some nonlinear PDE. namely, the nonlinear Pochhammer- Chree equation, the nonlinear generalized Zakharov-Kuznetsov equation, the nonlinear PDE describing the nonlinear dynamics of MTs as nanobioelectronics transmission lines, the nonlinear PDE describing the nonlinear dynamics of radial dislocations in MTs, the nonlineae PDE governing wave propagation in nonlinear low-pass electrical transmission lines and the nonlinear PDE describing pulse narrowing nonlinear transmission lines. The exact traveling wave solutions for these nonlinear PDEs include hyperbolic (kink and anti- kink solitons, bell and anti-bell solitary wave solutions), trigonometric (periodic solutions) and rational solutions. In chapter six, we apply two di¤erent methods to nd the exact traveling wave solu- tions of some nonlinear PDEs. Firstly, we nd the exact traveling wave solutions of the nonlinear fth-order Kaup-Kupershmidt equation (KK), the nonlinear fth-order Ito equation, the nonlinear fth-order Caudrey-Dodd-Gibbon equation (CDG), the nonlinear fth-order Lax equation and the nonlinear fth-order Sawada-Kotera equation (SK) us- ing the Kudryashov method. Secondly, we nd the exact traveling wave solutions of the nonlinear seventh-order Sawada-Kotera-Ito equation (SKI), the nonlinear seventh-order Kaup-Kupershmidt equation (KK) and the nonlinear seventh-order Lax equation using the modi ed Kudryashov method. The exact traveling wave solutions for these nonlinear PDEs include bell and anti-bell soliton wave solutions and symmetrical hyperbolic Lucas functions solutions. Finally, comparison between our recent results and the well-known results is given. We end this abstract with the remark that, we have published tweleve papers from this thesis, see [20,21,24,46,47,48,49,60,61,75,76,121]. |