Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 078, 12 pages      nlin.SI/0512016v2      http://dx.doi.org/10.3842/SIGMA.2006.078

Integrable Hierarchy of Higher Nonlinear Schrödinger Type Equations

Anjan Kundu
Saha Institute of Nuclear Physics, Theory Group & Centre for Applied Mathematics & Computational Science, 1/AF Bidhan Nagar, Calcutta 700 064, India

Received August 14, 2006, in final form October 17, 2006; Published online November 10, 2006

Abstract
Addition of higher nonlinear terms to the well known integrable nonlinear Schrödinger (NLS) equations, keeping the same linear dispersion (LD) usually makes the system nonintegrable. We present a systematic method through a novel Eckhaus-Kundu hierarchy, which can generate higher nonlinearities in the NLS and derivative NLS equations preserving their integrability. Moreover, similar nonlinear integrable extensions can be made again in a hierarchical way for each of the equations in the known integrable NLS and derivative NLS hierarchies with higher order LD, without changing their LD.

Key words: NLSE & DNLSE; higher nonlinearity; linear dispersion preservation; integrable Eckhaus-Kundu hierarchy.

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References

  1. Agarwal G., Nonlinear fiber optics, Elsevier, 2001, p. 50.
  2. Johnson R.S., On the modulation of water waves in the neighbourhood of kh 1.363, Proc. Roy. Soc. London Ser. A, 1977, V.357, 131-141.
  3. Benney D.J., A general theory for interactions between short and long waves, Studies in Appl. Math., 1976/77, V.57, 81-94.
  4. Kakutani T., Michihiro K., Marginal state of modulational instability-mode of Benjamin Feir instability, J. Phys. Soc. Japan, 1983, V.52, 4129-4137.
  5. Parkes E.J., The modulation of weakly non-linear dispersive waves near the marginal state of instability, J. Phys. A: Math. Gen., 1987, V.20, 2025-2036.
  6. Ndohi R., Kofane T.C., Solitary waves in ferromagnetic chains near the marginal state of instabilit, Phys. Lett. A, 1991, V.154, 377-380.
  7. Pelap F.B., Faye M.M., Solitonlike excitations in a one-dimensional electrical transmission line, J. Math. Phys., 2005, V.46, 033502, 10 pages.
  8. Sakovich S.Yu., Integrability of the higher order NLS revisited, nlin.SI/9906012.
  9. Kindyak A.S., Scott M.M., Patton C.E., Theoretical analysis of nonlinear pulse propagation in ferrite-dielectric-metal structures based on the nonlinear Schrödinger equation with higher order terms, J. Appl. Phys., 2003, V.93, 4739-4745.
  10. Zarmi Y., Perturbed NLS and asymptotic integrability, nlin.SI/0511057.
  11. Kundu A., Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations, J. Math. Phys., 1984, V.25, 3433-3438.
  12. Calogero F., Eckhaus W., Nonlinear evolution equations, rescalings, model PDEs and their integrability. I, Inverse Problems, 1987, V.3, 229-262.
  13. Clarkson P.A., Cosgrove C.M., Painlevé analysis of the nonlinear Schrödinger family of equations, J. Phys. A: Math. Gen., 1987, V.20, 2003-2024.
  14. Kakei S., Sasa N. Satsuma J., Bilinearization of a generalized derivative nonlinear Schrödinger equation, J. Phys. Soc. Japan, 1995, V.64, 1519-1523, solv-int/9501005.
  15. Feng Z., Wang X., Explicit exact solitary wave solutions for the Kundu equation and the derivative Schrödinger equation, Phys. Scripta, 2001, V.64, 7-14.
  16. Shen L.Y., Some algebraic properties of c-integrable nonlinear equation II-Eckhaus-Kundu equation and Thomas equation, Preprint, Univ. Sc. Tech. China, Hefei, China, 1989.
  17. Shen L.Y., Symmetries and constants of motion of integrable systems, in Symmetries & Singularity Structures, Spinger, 1990, 27-41.
  18. Conte R., Musette M., The Painlevé methods, in Classical and Quantum Nonlinear Integrable Systems, Bristol, IOP Publ., 2003, 39-63.
  19. Chen H.H., Lee Y.C., Liu C.S., Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scripta, 1979, V.20, 490-492.
  20. Gerdjikov V.S., Ivanov M.I., The quadratic bundle of general form and the nonlinear evolution equations: hierarchies of Hamiltonian structures, JINP Preprint E2-82-595, Dubna, 1982, 16 pages.
  21. Mikhailov A.V., Shabat A.B., Yamilov R.I., Extension of the module of invertible transformations. Classification of integrable systems, Comm. Math. Phys., 1988, V.115, 1-19.
  22. Mikhailov A.V., Shabat A.B., Yamilov R.I., A symmetric approach to the classification of nonlinear equations. Complete lists of integrable systems, Uspekhi Mat. Nauk, 1987, V.42, 3-53.
  23. Ablowitz M., Segur H., Solitons and the inverse scattering transform, Philadelphia, SIAM, 1981, p. 54.
  24. Novikov S.P. (Editor), Theory of solitons, Moscow, Nauka, 1980, p. 76 (in Russian).
  25. Kaup D.J., Newell A.C., An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 1978, V.19, 798-801.

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