Anjan Biswas*, M. Mirzazadeh, Mostafa Eslami, Daniela Milovic and Milivoj Belic Solitons in Optical Metamaterials by Functional Variable Method and First Integral Approach Abstract: This paper addresses soliton solutions in opti- cal metamaterials by functional variable method and first integral approach. These integration schemes lead to bright and singular 1-soliton solution. Topological soliton solution is also revealed. Additionally, as a byproduct, singular periodic solutions and continuous waves are ob- tained. The corresponding constraint conditions are also listed for these variety of solutions to exist. Finally, a nu- merical simulation of the bright 1-soliton solution is given. Keywords: solitons, metamaterials, integrability PACS ® (2010). 02.30.Ik, 02.30.Jr, 42.65.Tg DOI 10.1515/freq-2014-0050 Received April 2, 2014. 1 Introduction Solitons in optical metamaterials is one of the current and demanding topics of research. This rich field in non- linear optics receives a lot of attention from research and industrial applications perspectives. Optical soliton mole- cules form the basis of data transmission across trans- continental and trans-oceanic distances. Today, study of solitons in optical metamaterials is another promising lead in this direction. The integrability aspect of the model equation, that studies these solitons in optical metamaterials, is the focus of this paper. While there are several integration schemes that are studied at present times, there are two efficient algorithms that will be studied in this paper. They are functional variable method and first integral approach. These tools will lead to the extraction of bright and singular 1-soliton solutions to the model. Addition- ally, a couple of other nonlinear wave solutions will fall out that are not studied in the context of nonlinear op- tics. They are continuous wave solutions and singular periodic solutions. The constraint conditions are also dis- played for the existence of these waves. 2 Overview of integration algorithms This section will introduce the two integration architec- tures that will be implemented in this paper in order to obtain the optical soliton as well as other solutions to the governing equations. The following two subsections give the overview. 2.1 First integral approach One of the most effective direct methods to develop the traveling wave solution of nonlinear partial differential equations (NLPDEs) is the first integral method [12]. This method has been successfully applied to obtain exact solutions for a variety of NLPDEs [1, 2, 3, 14, 15, 17]. Dif- ferent from other traditional methods, the first integral method has many advantages, which is mainly embodied in that it could avoid a great deal of complicated and te- dious calculation and provide more exact and explicit traveling solitary solutions with high accuracy. Tascan et al. [17] summarized the main steps for using the first integral method, as follows: Step 1: Suppose a NLPDE Pðu; u t ; u x ; u tt ; u xt ; u xx ; ...Þ¼ 0; ð1Þ can be converted to an ODE QðU ; ωU 0 ; kU 0 ; ω 2 U 00 ; kωU 00 ; k 2 U 00 ; ...Þ¼ 0; ð2Þ Frequenz 2014; 68(11–12): 525–530 DE GRUYTER *Corresponding author: Anjan Biswas: Department of Mathematical Sciences, Delaware State University, Dover, USA; Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah-21589, Saudi Arabia. E-mail: biswas.anjan@gmail.com M. Mirzazadeh: Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar-Vajargah, Iran Mostafa Eslami: Department of Mathematics, Faculty of Sciences, University of Mazandaran, Babolsar, Iran Daniela Milovic: Department of Telecommunications, University of Nis, 18000 Niš, Serbia Milivoj Belic: Science Program, Texas A & M University at Qatar, Doha, Qatar Authenticated | biswas.anjan@gmail.com author's copy Download Date | 11/6/14 4:32 AM