The stationary equations of a coupled nonlinear Schrodinger system

The integrable coupled nonlinear Schrodinger (CNLS) equations under periodi c boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases [M.G. Forest, D.W. McLaughlin, D. Mura ki, O.C. Wright, Non-focussing instabilities in coupled, Integrable nonline ar Schrodinger PDEs, in preparation; D.J. Muraki, O.C. Wright, D.W. McLaugh lin, Birefringent optical fibers: Modulational instability in a near-integr able system, Nonlinear Processes in Physics: Proceedings of III Postdam-V K iev Workshop, 1991, pp. 242-245; O.C, Wright, Modulational stability in a d efocussing coupled nonlinear Schrodinger system, Physica D 82 (1995) 1-10], whereas the scalar NLS equation is linearly unstable only in the focussing case [M.G. Forest, J.E. Lee, Geometry and modulation theory for the period ic Schrodinger equation, in: Dafermas et al. (Eds.), Oscillation Theory, Co mputation, and Methods of Compensated Compactness, I.M.A, Math. Appl. 2 (19 86) 35-70]. These instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced Duffing oscilla tor [Y. Li, D.W. McLaughlin, Morse and Melnikov functions for NLS pde's, Co mmun. Math. Phys. 162 (1994) 175-214; D.W. McLaughlin, E.A. Overman, Whiske red tori for integrable Pde's: Chaotic behaviour in near integrable Pde's, in: Keller et al. (Eds.), Surveys in Applied Mathematics, vol. 1, Chapter 2 , Plenum Press, New York, 1995]. The homoclinic orbits and the near homocli nic tori that are connected to the unstable wave trains of the NLS and the CNLS reside in the finite-dimensional phase space of certain stationary equ ations [S.P Novikov, Funct. Anal. Prilozen, 8 (3) (1974) 54-66] of the infi nite hierarchy of integrable commuting flows. The correct stationary equati ons must be matched to the unstable torus through the analytic structure of the spectral curves [O.C, Wright, Near homoclinic orbits of the focussing nonlinear Schrodinger equation, preprint]. Thus, in this paper, the station ary equations of the CNLS are derived and the analytic structure of the tri gonal spectral curve is examined, providing a basis for further study of th e near homoclinic orbits of the CNLS system. (C) 1999 Elsevier Science B.V. All rights reserved.