Coupled periodic waves with opposite dispersions in a nonlinear optical fiber

S. C. Tsang, Nakkeeran Kaliyaperumal, B. A. Malomed, K. W. Chow

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)


Using the Hirota's method and elliptic theta-functions, we obtain three families of exact periodic (cnoidal) wave solutions for two nonlinear Schrodinger (NLS) equations coupled by XPM (cross-phase-modulation) terms, with a ratio a 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 sigma > 1 carry over into inverted bound states of bright and dark solitons in the normal- and anomalous-GVD modes (known as "symbiotic solitons"), while the infinite-period solution with sigma < 1 is an uninverted bound state (also an unstable one). The case of sigma = 2 is of direct interest to fiber-optic telecommunications, 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 sigma 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 sigma > 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 sigma = 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 GVD is weaker than the normal GVD in the mate channel. When the instability sets in, it first initiates quasi-reversible modulations, and only at a later stage the wave pattern decays into a "turbulent" state. (c) 2004 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)117-128
Number of pages11
JournalOptics Communications
Publication statusPublished - 2005


  • optical fiber
  • coupled nonlinear Schrodinger (NLS) equations
  • periodic solutions
  • Hirota method


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