## Complex Toda lattice and its application to the theory of interacting optical solitons

JOSA A, Vol. 15, Issue 5, pp. 1450-1458 (1998)

http://dx.doi.org/10.1364/JOSAA.15.001450

Acrobat PDF (261 KB)

### Abstract

A class of linear and nonlinear dynamical problems that arise when studying the modulation of trains of nearly identical soliton pulses of the nonlinear Schrödinger equation is introduced. In the simplest case the dynamics of the nonlinear Schrödinger equation can be reduced to an equation that is a complex extension of the integrable Toda lattice equation, so that the latter asymptotically models the former in the case of large intersoliton separations.

© 1998 Optical Society of America

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**Citation**

J. M. Arnold, "Complex Toda lattice and its application to the theory of interacting optical solitons," J. Opt. Soc. Am. A **15**, 1450-1458 (1998)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-5-1450

Sort: Year | Journal | Reset

### References

- J. M. Arnold, “Soliton pulse position modulation,” Proc. Inst. Electr. Eng. 140, 359–366 (1993).
- J. M. Arnold, “Digital pulse position modulation of optical fibre solitons,” Opt. Lett. 21, 31–33 (1996).
- J. M. Arnold, “Stability of nonlinear pulse trains on optical fibres,” in Proceedings of Electromagnetic Theory Symposium (International Union of Radio Science, Gent, Belgium, 1995), pp. 553–555.
- V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, and E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
- V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
- V. Karpman and Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
- J. M. Arnold, “Stability theory for periodic pulse train solutions of the nonlinear Schrödinger equation,” IMA J. Appl. Math. 52, 123–140 (1994).
- A. Hasegawa and Y. Kodama, Optical Fibre Solitons (Cambridge U. Press, Cambridge, UK, 1994).
- G. B. Whitham, Linear and Nonlinear Waves (Academic, Orlando, Fla., 1974).
- K. A. Gorshkov and L. A. Ostrovski, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
- D. Anderson and M. Lisak, “Bandwidth limits due to interpulse interaction in optical soliton communication systems,” Opt. Lett. 11, 174–176 (1986).
- L. D. Fadeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
- J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons,” Opt. Lett. 11, 665–667 (1986).
- V. E. Zhakarov and A. B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
- J. Satsuma and N. Yajima, “Initial value problem of one- dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
- E. J. Hinch, Perturbation Methods (Cambridge U. Press, Cambridge, UK, 1991).
- S. Manakov, “Complete integrability and stochastization of discrete dynamical systems,” Sov. Phys. JETP 40, 269–274 (1974).
- H. Flaschka and D. McLaughlin, “Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions,” Prog. Theor. Phys. 55, 438–456 (1976).
- H. Flaschka, “The Toda lattice. I. Existence of integrals,” Phys. Rev. B 9, 1924–1925 (1974).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

OSA is a member of CrossRef.