JOURNAL OF OPTOELECTRONICS AND ADVANCED MATERIALS Vol. 16, No. 5-6, May - June 2014, p. 750 - 758 Optical solitons in multiple-core couplers A. A. ALSHAERY a , E. M. HILAL a , M. A. BANAJA a , SADAH A. ALKHATEEB a , LUMINITA MORARU b , ANJAN BISWAS c,d . a Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia b Department of Chemistry, Physics and Environment, University Dunarea de Jos Galati, 111 Domneasca Street, 800201 Galati, Romania c Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA d Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia This paper obtains 1-soliton solutions in multiple-core nonlinear directional couplers. Bright, dark and singular soliton solutions are obtained. The necessary constraint conditions are exhibited. Coupling with nearest neighbors as well as all neighbors are considered. There are five types of nonlinear media that are studied. They are Kerr law, power law, parabolic law, dual-power law and log law. (Received May 7, 2014; accepted May 15, 2014) Keywords: Multiple-core, Integrability, Constraints 1. Introduction The dynamics of solitons in nonlinear directional couplers (NLDC) has been studied for the past few years in the context of nonlinear optical fibers [1-30]. There are several forms of results that are being produced and they are mostly numerical simulations However, what is not visible is the analytical aspect of NLDC. Analytical results are truly missing from this literature. Therefore, it is important to address this aspect so that the gap can be filled in. This paper addresses the analytical aspects of optical solitons in multiple-core optical couplers where coupling is with nearest neighbors as well as with all neighbors. Bright, dark and singular optical soliton solutions will be retrieved along with several necessary constraint conditions on the soliton parameters. There are five forms of nonlinear media that will be studied in this context. They are Kerr law, power law, parabolic law, dual-power law and log law. The results will be extremely helpful in the study of optical routing as well as switching and other studies. The detailed study will now be conducted in the following two sections. 2. Coupling with nearest neighbors The governing equation for multiple-core couplers is given by [1, 9, 10, 30] ] 2 [ ) ( ) 1 ( ) ( ) 1 ( ) ( 2 ) ( ) ( ) (         l l l l l l xx l t q q q K q q bF aq iq (1) where N l   1 . Equation (1) represents the general model for optical couplers where coupling with nearest neighbors is considered. In (1), a is the coefficient of group velocity dispersion (GVD) and b is the coefficient of nonlinearity. The functional F represents non-Kerr law nonlinear media that will be studied in details in the next five subsections. On the right hand side, K represents coupling coefficient. It must be noted that the general case of optical couplers with the inclusion of spatio-temporal dispersion is already studied earlier [30]. This paper however considers only GVD. In order to address this model for the five forms of nonlinear media, the initial hypothesis is taken to be ) , ( ) ( ) , ( ) , ( t x i l l e t x P t x q   (2) where the amplitude component of soliton is P l (x, t) while the phase component is defined as         t x t x ) , ( (3) Here, κ is the soliton frequency, while ω is the wave number and θ is the phase constant. After substituting hypothesis (2) into (1) while utilizing (3), the resulting expression is split into real and imaginary components. The imaginary part gives the speed of the soliton as    a v 2 (4) The speed of the soliton stays the same for any kind of nonlinearity as well as for all types of nonlinear media and all kinds of solitons. Next, the real part implies 0 ] 2 [ ) ( ) ( 1 1 2 2 2 2               l l l l l l l P P P K P P bF x P a a P (5) It is this real part equation that will be further analyzed in the next five subsections based on the types of nonlinearity and types of solitons.