×

On the non-chiral intermediate long wave equation. (English) Zbl 1496.35308

Summary: We study integrability properties of the non-chiral intermediate long wave equation recently introduced by the authors as a parity-invariant variant of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) derive a Bäcklund transformation, (d) use, separately, the Bäcklund transformation and the Lax representation to obtain an infinite number of conservation laws.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q51 Soliton equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Joseph, R. I., Solitary waves in a finite depth fluid, J. Phys. A: Math. Gen., 10, L225-L227 (1977) · Zbl 0369.76018 · doi:10.1088/0305-4470/10/12/002
[2] Kodama, Y.; Satsuma, J.; Ablowitz, M. J., Nonlinear intermediate long-wave equation: analysis and method of solution, Phys. Rev. Lett., 46, 687-690 (1981) · doi:10.1103/physrevlett.46.687
[3] Kodama, Y.; Ablowitz, M. J.; Satsuma, J., Direct and inverse scattering problems of the nonlinear intermediate long wave equation, J. Math. Phys., 23, 564-576 (1982) · Zbl 0489.35004 · doi:10.1063/1.525393
[4] Matsuno, Y., Exact multi-soliton solution for nonlinear waves in a stratified fluid of finite depth, Phys. Lett. A, 74, 233-235 (1979) · doi:10.1016/0375-9601(79)90779-5
[5] Satsuma, J.; Ablowitz, M. J.; Kodama, Y., On an internal wave equation describing a stratified fluid with finite depth, Phys. Lett. A, 73, 283-286 (1979) · doi:10.1016/0375-9601(79)90534-6
[6] Berntson, B. K.; Langmann, E.; Lenells, J., Non-chiral intermediate long-wave equation and inter-edge effects in narrow quantum Hall systems, Phys. Rev. B, 102 (2020) · doi:10.1103/physrevb.102.155308
[7] Chen, H. H.; Lee, Y. C.; Pereira, N. R., Algebraic internal wave solitons and the integrable Calogero-Moser-Sutherland N-body problem, Phys. Fluids, 22, 187-188 (1979) · doi:10.1063/1.862457
[8] Foursov, M. V., Towards the complete classification of homogeneous two-component integrable equations, J. Math. Phys., 44, 3088-3096 (2003) · Zbl 1062.37068 · doi:10.1063/1.1580998
[9] Karasu, A., Painlevé classification of coupled Korteweg-de Vries systems, J. Math. Phys., 38, 3616-3622 (1997) · Zbl 0882.58026 · doi:10.1063/1.532056
[10] Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear equations, Stud. Appl. Math., 139, 7-59 (2017) · Zbl 1373.35281 · doi:10.1111/sapm.12153
[12] Abanov, A. G.; Bettelheim, E.; Wiegmann, P., Integrable hydrodynamics of Calogero-Sutherland model: bidirectional Benjamin-Ono equation, J. Phys. A: Math. Theor., 42 (2009) · Zbl 1168.37022 · doi:10.1088/1751-8113/42/13/135201
[13] Calogero, F., Classical Many-Body Problems Amenable to Exact Treatments (2001), Berlin: Springer, Berlin · Zbl 1011.70001
[14] Pelinovsky, D., Intermediate nonlinear Schrödinger equation for internal waves in a fluid of finite depth, Phys. Lett. A, 197, 401-406 (1995) · doi:10.1016/0375-9601(94)00991-w
[15] Matsuno, Y., Multiperiodic and multisoliton solutions of a nonlocal nonlinear Schrödinger equation for envelope waves, Phys. Lett. A, 278, 53-58 (2000) · doi:10.1016/s0375-9601(00)00757-x
[16] Pelinovsky, D. E.; Grimshaw, R. H J., A spectral transform for the intermediate nonlinear Schrödinger equation, J. Math. Phys., 36, 4203-4219 (1995) · Zbl 0845.35119 · doi:10.1063/1.530956
[17] Matsuno, Y., Exactly solvable eigenvalue problems for a nonlocal nonlinear Schrödinger equation, Inverse Problems, 18, 1101-1125 (2002) · Zbl 1087.35531 · doi:10.1088/0266-5611/18/4/311
[18] Matsuno, Y., A Cauchy problem for the nonlocal nonlinear Schrödinger equation, Inverse Problems, 20, 437-445 (2004) · Zbl 1081.35105 · doi:10.1088/0266-5611/20/2/008
[19] Zhang, Y-J, A class of integro-differential equations constrained from the KP hierarchy, J. Phys. A: Math. Gen., 27, 8149-8160 (1994) · Zbl 0838.45005 · doi:10.1088/0305-4470/27/24/022
[20] Ablowitz, M. J.; Fokas, A. S.; Satsuma, J.; Segur, H., On the periodic intermediate long wave equation, J. Phys. A: Math. Gen., 15, 781-786 (1982) · Zbl 0496.45013 · doi:10.1088/0305-4470/15/3/017
[21] Matsuno, Y., The Benjamin-Ono equation, Bilinear Transformation Method, 47-95 (1984), Amsterdam: Elsevier, Amsterdam
[22] Gérard, P.; Grellier, S., The cubic Szegö equation, Ann. Sci. École Norm. Sup., 43, 761-810 (2010) · Zbl 1228.35225 · doi:10.24033/asens.2133
[23] Zhou, T.; Stone, M., Solitons in a continuous classical Haldane-Shastry spin chain, Phys. Lett. A, 379, 2817-2825 (2015) · Zbl 1349.37076 · doi:10.1016/j.physleta.2015.09.014
[24] Lenzmann, E.; Schikorra, A., On energy-critical half-wave maps into \(####\), Invent Math., 213, 1-82 (2018) · Zbl 1411.35208 · doi:10.1007/s00222-018-0785-1
[25] Berntson, B. K.; Klabbers, R.; Langmann, E., Multi-solitons of the half-wave maps equation and Calogero-Moser spin-pole dynamics, J. Phys. A: Math. Theor., 53 (2020) · Zbl 1519.81258 · doi:10.1088/1751-8121/abb167
[26] Scoufis, G.; Cosgrove, C. M., An application of the inverse scattering transform to the modified intermediate long wave equation, J. Math. Phys., 46 (2005) · Zbl 1111.35066 · doi:10.1063/1.1996830
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.