Dynamics of solitons in plasmas for the complex KdV equation with power law nonlinearity Anjan Biswas a, * , Essaid Zerrad b , Arjuna Ranasinghe c a Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA b Department of Physics and Pre-Engineering, Delaware State University, Dover, DE 19901-2277, USA c Department of Mathematics, Alabama A & M University, Normal, AL-35762, USA article info Keywords: Solitons Perturbation Dark solitons abstract This paper obtains the 1-soliton solution of the complex KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The soliton per- turbation theory for this equation is developed and the soliton cooling is observed for bright solitons. Finally, the dark soliton solution is also obtained for this equation. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction The complex KdV (cKdV) equation appears in many areas of Physics and Mathematics [1–10]. In particular, they show up in Nonlinear Optics in the context of embedded solitons [8] and also in the area of Plasma Physics [5,7]. This equa- tion is basically a byproduct of the nonlinear Schrödinger’s equation that studies the propagation of solitons through optical fibers. The Sasa–Satsuma equation that is studied in the context of propagation of solitons through optical fibers, also leads to the cKdV equation [2]. Furthermore, there are various forms of multi-component generalizations of this equation [2]. In this paper, the concentration is going to be on the scalar version of the cKdV equation with power law nonlinearity. There are various methods of integrating these nonlinear evolution equations. The most powerful technique is the clas- sical Inverse Scattering Transform (IST) that is the nonlinear analog of Fourier Transform which is used for solving linear par- tial differential equations. There are various modern methods of integrability that are developed mainly in the last couple of decades to integrate many other nonlinear evolution equations which cannot be integrated by IST, although it is a powerful technique. Some of these modern methods of integrability are known as Adomian decomposition method, exponential func- tion method, He’s variational principle, sub-equation method just to name a few. However, one needs to be careful when applying these methods as it could lead to erroneous results [6]. There will be one such method that will be used to carry out the integration of the cKdV equation with power law non- linearity. This paper will study the cKdV equation with power law nonlinearity and the integration will be carried out using the soliton ansatz method. Then the soliton perturbation theory will be developed. A few exemplary perturbation terms will be taken and the adiabatic parameter dynamics of the soliton parameters will be obtained and embedded solitons will be introduced. Finally, the topological solitons will be touched upon. 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.025 * Corresponding author. E-mail address: biswas.anjan@gmail.com (A. Biswas). Applied Mathematics and Computation 217 (2010) 1491–1496 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc