Optical Solitons with Power Law Nonlinearity and Hamiltonian Perturbations: An Exact Solution
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Optical Solitons with Power Law Nonlinearity and Hamiltonian Perturbations: An Exact Solution Amarendra K. Sarma · Manirupa Saha · Anjan Biswas
Received: 17 March 2010 / Accepted: 30 June 2010 / Published online: 17 July 2010 © Springer Science+Business Media, LLC 2010
Abstract This paper studies the optical solitons with power law nonlinearity in presence of Hamiltonian perturbation terms. The perturbation terms that are taken into account are inter-modal dispersion, third order dispersion, self-steepening term and nonlinear dispersion. An exact 1-soliton solution is obtained by the solitary wave ansatz. Both bright and dark optical soliton solutions are obtained. The domain restrictions have also been identified in the process. Keywords Optical solitons · Dark solitons · Integrability · Exact solution AMS 2000 Subject Classifications 78A60 · 37 K10 · 35Q51 · 35Q55 OCIS 060.2310 · 060.4510 · 060.5530 · 190.3270 · 190.4370 1 Introduction Optical solitons is one of the major areas of research in the field of Nonlinear Optics. This area of research has made an enormous progress especially in the past couple of decades. There has been an overwhelming number of publications that appeared during this time frame [1–25]. One of the important
A. K. Sarma · M. Saha Department of Physics, Indian Institute of Technology, Guwahati, Guwahati 781039, India A. Biswas (B) Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications, Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA e-mail: [email protected]
J Infrared Milli Terahz Waves (2010) 31:1048–1056
1049
aspects of this area is the issue of integrability in presence of perturbation terms especially for non-Kerr law nonlinearity [1]. Although many numerical simulations are done, the exact soliton solution always is a very useful piece of information that supplements the numerics [17, 18]. In this paper, one such non-Kerr law nonlinearity will be considered. This is the power law nonlinearity. Thus, the governing equation, namely the nonlin¨ ear Schrdinger’s equation (NLSE) with power law nonlinearity will be studied in presence of Hamiltonian perturbation terms. An exact 1-soliton solution will be obtained. The domain restrictions of the soliton parameters will be identified and the constraint relation between the perturbation coefficients will also be obtained. There are various methods of carrying out the integration of the NLSE. Some of these techniques are the Inverse Scattering Transform [1], Lie symmetry method [6], He’s semi-inverse variational principle [4], exponential function method, G /G method, Adomian decomposition method [24], variational iteration method [25], collective variables approach [21] and many more. In this paper, one such method of integration will be used to carry out the integration of the governing NLSE to obtain the exact 1-soliton solution. It is the method of soliton ansatz [2].
2 Governing equation The governing NLSE, in dimensionless form, for
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