The Stationary Equations of a Coupled Nonlinear Schroedinger System
The integrable coupled nonlinear Schro¨dinger (CNLS) equations under periodic boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases [M.G. Forest, D.W. McLaughlin, D. Muraki, O.C. Wright, Non-focussing instabilities in coupled, Integrable nonlinear Schro¨dinger PDEs, in preparation; D.J. Muraki, O.C. Wright, D.W. McLaughlin, Birefringent optical fibers: Modulational instability in a near-integrable system, Nonlinear Processes in Physics: Proceedings of III Postdam-V Kiev Workshop, 1991, pp. 242–245; O.C. Wright, Modulational stability in a defocussing coupled nonlinear Schro¨dinger 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 periodic Schro¨dinger equation, in: Dafermas et al. (Eds.), Oscillation Theory, Computation, and Methods of Compensated Compactness, I.M.A. Math. Appl. 2 (1986) 35–70]. These instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced Duffing oscillator [Y. Li, D.W. McLaughlin, Morse and Melnikov functions for NLS pde’s, Commun. Math. Phys. 162 (1994) 175–214; D.W. McLaughlin, E.A. Overman, Whiskered 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 homoclinic 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 equations [S.P. Novikov, Funct. Anal. Prilozen, 8 (3) (1974) 54–66] of the infinite hierarchy of integrable commuting flows. The correct stationary equations 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 Schro¨dinger equation, preprint]. Thus, in this paper, the stationary equations of the CNLS are derived and the analytic structure of the trigonal spectral curve is examined, providing a basis for further study of the near homoclinic orbits of the CNLS system.
Lax pair, Riemann surface, Integrable system, Quasiperiodic solutions, Lie algebra
Wright, O. C. (1999). The Stationary Equations of a Coupled Nonlinear Schroedinger System. Physica D, 126, 275-289.