Integrable Hierarchy of Higher Nonlinear Schrödinger Type Equations

Anjan Kundu

Symmetry, integrability and geometry: methods and applications, Tome 2 (2006)

Résumé

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.

Détail

Keywords: NLSE & DNLSE; higher nonlinearity; linear dispersion preservation; integrable Eckhaus–Kundu hierarchy.

Bibliographie

[1] Agarwal G., Nonlinear fiber optics, Elsevier, 2001, p. 50 | Zbl

[2] Johnson R. S., “On the modulation of water waves in the neighbourhood of $kh\approx 1.363$”, Proc. Roy. Soc. London Ser. A, 357 (1977), 131–141 | DOI | MR | Zbl

[3] Benney D. J., “A general theory for interactions between short and long waves”, Studies in Appl. Math., 57 (1976/77), 81–94 | MR

[4] Kakutani T., Michihiro K., “Marginal state of modulational instability-mode of Benjamin Feir instability”, J. Phys. Soc. Japan, 52 (1983), 4129–4137 | DOI

[5] Parkes E. J., “The modulation of weakly non-linear dispersive waves near the marginal state of instability”, J. Phys. A: Math. Gen., 20 (1987), 2025–2036 | DOI

[6] Ndohi R., Kofane T. C., “Solitary waves in ferromagnetic chains near the marginal state of instabilit”, Phys. Lett. A, 154 (1991), 377–380 | DOI

[7] Pelap F. B., Faye M. M., “Solitonlike excitations in a one-dimensional electrical transmission line”, J. Math. Phys., 46 (2005), 033502, 10 pp., ages | DOI | MR | Zbl

[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., 93 (2003), 4739–4745 | DOI

[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., 25 (1984), 3433–3438 | DOI | MR

[12] Calogero F., Eckhaus W., “Nonlinear evolution equations, rescalings, model PDEs and their integrability. I”, Inverse Problems, 3 (1987), 229–262 | DOI | MR | Zbl

[13] Clarkson P. A., Cosgrove C. M., “Painlevé analysis of the nonlinear Schrödinger family of equations”, J. Phys. A: Math. Gen., 20 (1987), 2003–2024 | DOI | MR | Zbl

[14] Kakei S., Sasa N., Satsuma J., “Bilinearization of a generalized derivative nonlinear Schrödinger equation”, J. Phys. Soc. Japan, 64 (1995), 1519–1523 ; solv-int/9501005 | DOI | MR | Zbl

[15] Feng Z., Wang X., “Explicit exact solitary wave solutions for the Kundu equation and the derivative Schrödinger equation”, Phys. Scripta, 64 (2001), 7–14 | DOI | MR | Zbl

[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”, Symmetries Singularity Structures, Spinger, 1990, 27–41

[18] Conte R., Musette M., “The Painlevé methods”, Classical and Quantum Nonlinear Integrable Systems, IOP Publ., Bristol, 2003, 39–63

[19] Chen H. H., Lee Y. C., Liu C. S., “Integrability of nonlinear Hamiltonian systems by inverse scattering method”, Phys. Scripta, 20 (1979), 490–492 | DOI | MR | Zbl

[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 pp., ages | Zbl

[21] Mikhailov A. V., Shabat A. B., Yamilov R. I., “Extension of the module of invertible transformations. Classification of integrable systems”, Comm. Math. Phys., 115 (1988), 1–19 | DOI | MR

[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, 42:4(256) (1987), 3–53 | MR

[23] Ablowitz M., Segur H., Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981, p. 54 | MR | Zbl

[24] Novikov S. P. (ed.), Theory of solitons, Nauka, Moscow, 1980, p. 76 (in Russian) | MR

[25] Kaup D. J., Newell A. C., “An exact solution for a derivative nonlinear Schrödinger equation”, J. Math. Phys., 19 (1978), 798–801 | DOI | MR | Zbl