%0 Journal Article
%T A Modified Approach to the New Solutions of Generalized mKdV Equation Using -Expansion
%A Yu Zhou
%A Ying Wang
%J ISRN Applied Mathematics
%D 2014
%R 10.1155/2014/856396
%X The modified -expansion method is applied for finding new solutions of the generalized mKdV equation. By taking an appropriate transformation, the generalized mKdV equation is solved in different cases and hyperbolic, trigonometric, and rational function solutions are obtained. 1. Introduction The evolutions of the physical, engineering, and other systems always behave nonlinearly; hence many nonlinear evolution equations have been introduced to interpret the phenomena. Many kinds of mathematical methods have been established to investigate the solutions of those nonlinear evolution equations both numerically and asymptotically, while the exact solutions are of particular interests. In recent decades, with the rapid progress of computation methods, many effective calculating approaches have been developed, for example, the tanh-coth expansion [1, 2], -expansion [3, 4], Painlev¨¦ expansion [5], Jacobi elliptic function method [6], Hirota bilinear transformation [7], Backlund/Darboux transformation [8, 9], variational method [10], the homogeneous balance method [11], exp-function expansion [12], and so on. However, a unified approach to obtain the complete solutions of the nonlinear evolution equations has not been revealed. Within recent years, a new method called ( )-expansion [13] has been proposed for finding the traveling wave solutions of the nonlinear evolution equations. Many equations have been investigated and many solutions have been found using the method, including KdV equation, Hirota-Satsuma equation [13], coupled Boussinesq equation [14], generalized Bretherton equation [15], the mKdV equation [16], the Burgers-KdV equation, the Benjamin-Bona-Mahony equation [17], the Whitham-Broer-Kaup-like equation [18], the Kolmogorov-Petrovskii-Piskunov equation [19], KdV-Burgers equation [20], and Drinfeld-Sokolov-Satsuma-Hirota equation [21]. The mKdV equation, a modified version of the Korteweg-de Vries (KdV) equation, has been investigated extensively since Zabusky showed how this equation depict the oscillations of a lattice of particles connected by nonlinear springs as the Fermi-Pasta-Ulam (FPU) model [22¨C25]. Afterwards, this equation has been used to describe the evolution of internal waves at the interface of two layers of equal depth [26]. Generally, the KdV theory describes the weak nonlinearity and weak dispersion while, in the study of nonlinear optics, the complex mKdV equation has even been used to describe the propagation of optical pulses in nematic optical fibers when we go beyond the usual weakly nonlinear limit of Kerr medium [27].
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2014/856396/