Symmetry approach for a three coupled Schroedinger equation system A. Catania, M. Torrisi * , R. Tracinà Dipartimento di Matematica e Informatica, Viale A. Doria 6, 95125 Catania, Italy article info Keywords: Schroedinger equations Symmetries Exact solutions abstract We consider a system of three coupled cubic Schroedinger equations arising in some phys- ical and biomathematical problems. By using the Lie symmetry methods we get new clas- ses of exact solutions. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction The systems of N coupled cubic Schroedinger equations of the form iq m;t þ c m q m;xx þ 2 X N l¼1 a ml jq l j 2 q m ¼ 0; m ¼ 1; 2; ... ; N ð1:1Þ have been used to mainly describe the multimode propagation of electromagnetic waves in optical fiber. In the recent years they have been extensively studied for their increasing importance in several physical and mathematical problems. In par- ticular a lot of results have been put up for N ¼ 2. In this case it is possible to find not only many exact solutions, obtained by following different approaches, but several papers that have investigated about its integrability properties (see e.g. [1,2] and references inside). For N P 3 there are essentially results concerned with integrability property but the papers where exact solutions have been derived are scarce and these ones are mainly solitonic solutions [3]. In the special case N ¼ 3, this system is of considerable physical interest. In fact, in addition to optical fiber communica- tion, in the framework of biophysics the system can be used to study the propagation along the three spines of an alpha helix in protein [4,5]. In this paper, we study the system (1.1) for N ¼ 3 by following the classical Lie group analysis approach in order to get some special classes of exact solutions and show to the interested readers a tool to find new classes of solutions in the case under consideration. In fact the group analysis approach offers a methodological way to get exact solutions for nonlinear differential equations. To the best of our knowledge there are not symmetry studies that have been done concerning with a system of three coupled cubic Schroedinger equations. Several papers, instead, exist concerning with symmetry analysis of single Schroedinger equations and of systems of two coupled Schroedinger equations (see e.g. [6,7] and references therein). The plan of the paper is the following: in Section 2, after having obtained the infinitesimal generator of the symmetries, we rewrite the system under consideration by using the exponential form for the complex variables q m and transform the previous results in the new variables. In Section 3 by using the infinitesimal generator written in the new variables we derive some reduced systems and get solutions for them. Finally in Section 4 we write the finite form of the symmetry transforma- 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.07.006 * Corresponding author. E-mail addresses: catania@dmi.unict.it (A. Catania), torrisi@dmi.unict.it (M. Torrisi), tracina@dmi.unict.it (R. Tracinà). Applied Mathematics and Computation 204 (2008) 408–415 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc