Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations Houria Triki a , Abdul-Majid Wazwaz b, * a Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria b Department of Mathematics, Saint Xavier University, Chicago, IL 60655, United States article info Keywords: Multi-dimensional K(m, n) equation Soliton solution Conserved quantity Perturbation abstract We consider the nonlinear dispersive Kðm; nÞ equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech p function, we obtain exact bright soliton solutions for (2 + 1)- dimensional and (3 + 1)-dimensional Kðm; nÞ equations with the generalized evolution terms. The results are then generalized to multi-dimensional Kðm; nÞ equations in the pres- ence of the generalized evolution term. An extended form of the Kðm; nÞ equation with per- turbation term is investigated. Exact bright soliton solution for the proposed Kðm; nÞ equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction In recent years, there has been a growing interest in finding exact analytical solutions to nonlinear wave equations by using appropriate techniques. Particularly, the existence of soliton solutions for nonlinear models is of great importance be- cause of their potential application in many physics areas such as nonlinear optics, plasmas, fluid mechanics, condensed matter and many more. Solitons are defined as localized waves that propagate without change of its shape and velocity prop- erties and stable against mutual collisions [1]. It is necessary to note that the formation of this kind of pulses is due to perfect balance between nonlinearity and dispersion effects under specific conditions. Solitons are special solutions of a certain class of nonlinear evolution equations such as the nonlinear Schrödinger (NLS) equation, the KdV equation, and the sine-Gordon (sG) equation. In many practical physics problems, the resulting nonlinear wave equations of interest are non-integrable [2]. In some particular cases they may be close to an integrable one [2]. It is remarkable that non-integrability is not necessarily related to the nonlinear terms [3]. Higher order dispersions, for example, also can make the system to be non-integrable (while it remains Hamiltonian) [3]. The inverse scattering transform (IST) method, introduced by Gardner et al. (1967) [4], is a pow- erful tool to investigate the integrable systems. However, in the concrete application of the IST approach there is still some difficulties [5]. To apply the IST approach, the first step is to find a so-called ‘Lax pair’ [5]. Again, this method is applicable only to a few restricted cases in which a convergent solution is obtainable [6]. It is then essential to look for new techniques to find exact analytical solutions of partial differential equations of physical relevance [2]. In recent years, many significant methods have been established and developed to construct exact analytical solutions of nonlinear evolution equations. Among these methods, we can cite the hyperbolic tangent method [7], the coupled ampli- tude-phase formulation [8], Hirota bilinear method [9], the sub-ODE method [10,11], and other methods as well. 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.025 * Corresponding author. E-mail address: wazwaz@sxu.edu (A.-M. Wazwaz). Applied Mathematics and Computation 217 (2010) 1733–1740 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc