Coupled periodic waves with opposite dispersions in a nonlinear optical fiber S.C. Tsang a , K. Nakkeeran b, * , Boris A. Malomed c , K.W. Chow a a Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong b Photonics Research Center, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong c Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel Received 13 September 2004; received in revised form 17 December 2004; accepted 20 December 2004 Abstract Using the HirotaÕs method and elliptic h-functions, we obtain three families of exact periodic (cnoidal) wave solu- tions for two nonlinear Schro ¨ dinger (NLS) equations coupled by XPM (cross-phase-modulation) terms, with a ratio r of the XPM and SPM (self-phase-modulation) coefficients. Unlike the previous works, we obtain the solutions for the case when the coefficients of the group-velocity-dispersion (GVD) in the coupled equations have opposite signs. In the limit of the infinite period, the solutions with r > 1 carry over into inverted bound states of bright and dark sol- itons in the normal- and anomalous-GVD modes (known as ‘‘symbiotic solitons’’), while the infinite-period solution with r < 1 is an uninverted bound state (also an unstable one). The case of r = 2 is of direct interest to fiber-optic tele- communications, as it corresponds to a scheme with a pulse stream in an anomalous-GVD payload channel stabilized by a concomitant strong periodic signal in a mate normal-GVD channel. The case of arbitrary r may be implemented in a dual-core waveguide. To understand the stability of the coupled waves, we first analytically explore the modulational stability of CW (constant-amplitude) solutions, concluding that they may be completely stable for r P 1, provided that the absolute value of the GVD coefficient is smaller in the anomalous-GVD mode than in the normal-GVD one, and certain auxiliary conditions on the amplitudes are met. The stability of the exact cnoidal-wave solutions is tested in direct simulations. We infer that, while, strictly speaking, in the practically significant case of r = 2 all the solutions are unstable, in many cases the instability may be strongly attenuated, rendering the above-mentioned paired channels scheme usable. In particular, the instability is milder for a smaller period of the wave pattern, and/or if the anomalous 0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.12.042 * Corresponding author. Tel.: +852 2766 6197; fax: +852 2362 8439. E-mail addresses: ennaks@polyu.edu.hk (K. Nakkeeran), malomed@post.tau.ac.il (B.A. Malomed), kwchow@hkusua.hku.hk (K.W. Chow). Optics Communications 249 (2005) 117–128 www.elsevier.com/locate/optcom