×

Some models for the interaction of long and short waves in dispersive media. I: Derivation. (English) Zbl 1456.35157

Summary: It is universally accepted that the cubic, nonlinear Schrödinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Korteweg-de Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves, and such a system has been studied over the last few decades. However, questions about the validity of this system in the study of water waves were raised in a previous work of one of us where the analysis was presented using the fifth-order KdV as the starting point. These questions will now be settled unequivocally in a series of papers. In this first part, we show that the NLS-KdV system (or even the linear Schrödinger-KdV system) cannot be resulted from the full Euler equations formulated in the study of water waves. In the process of so doing, we also propose a few alternative models for describing the interaction of long and short waves.

MSC:

35Q31 Euler equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q53 KdV equations (Korteweg-de Vries equations)
35A15 Variational methods applied to PDEs
35B35 Stability in context of PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albert, J.; Bhattarai, S., Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Adv. Diff. Eqns., 18, 1129-1164 (2013) · Zbl 1290.35219
[2] Angulo Pava, J., Stability of solitary wave solutions for equations of short and long dispersive waves, Elec. Jour. Diff. Eqns., 72, 1-18 (2006) · Zbl 1110.35071
[3] Angulo Pava, J.; Matheus, C.; Pilod, D., Global well-posedness and nonlinear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system, Comm. Pure and Applied Anal., 8, 3, 815-844 (2009) · Zbl 1168.35430
[4] Appert, K.; Vaclavik, J., Dynamics of coupled solitons, Phys. Fluids, 20, 1845-1849 (1977) · Zbl 0368.76099 · doi:10.1063/1.861802
[5] Benney, DJ, A general theory for interactions between short and long waves, Studies Appl. Math., 56, 81-94 (1977) · Zbl 0358.76011 · doi:10.1002/sapm197756181
[6] Benney, DJ; Newell, AC, The propagation of nonlinear wave envelopes, Jour. Math. Phys., 46, 133-139 (1967) · Zbl 0153.30301 · doi:10.1002/sapm1967461133
[7] Bona, JL; Chen, M.; Saut, S-C, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: derivation and linear theory, J. Nonlinear Sci., 12, 283-318 (2002) · Zbl 1022.35044 · doi:10.1007/s00332-002-0466-4
[8] Bona, JL; Chen, M.; Saut, S-C, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II: the nonlinear theory, Nonlinearity, 17, 925-952 (2004) · Zbl 1059.35103 · doi:10.1088/0951-7715/17/3/010
[9] Bona, JL; McKinney, WR; Restrepo, JM, Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation, Jour. Nonlinear Sci., 10, 603-638 (2000) · Zbl 0972.35131 · doi:10.1007/s003320010003
[10] Bona, JL; Pritchard, WG; Scott, LR, An evaluation of a model equation for water waves, Phil. Trans. Roy. Soc. London A, 302, 457-510 (1981) · Zbl 0497.76023 · doi:10.1098/rsta.1981.0178
[11] Bona, J.L., Pritchard, W.G., Scott, L.R.: A comparison of solutions of two model equations for long waves, in Fluid Dynamics in Astrophysics and Geophysics (Chicago, Ill., 1981), Amer. Math. Soc., Providence, RI, (1983), 235-267 (1983) · Zbl 0534.76024
[12] Bona, JL; Smith, R., A model for the two-way propagation of water waves in a channel, Math. Proc. Cambridge Phil. Soc., 79, 167-182 (1976) · Zbl 0332.76007 · doi:10.1017/S030500410005218X
[13] Brinch-Nielsen, U.; Jonsson, IG, Fourth order evolution equations and stability analysis for Stokes waves on arbitrary water depth, Wave Motion, 8, 455-472 (1986) · Zbl 0625.76020 · doi:10.1016/0165-2125(86)90030-2
[14] Chen, L., Orbital stability of solitary waves of the nonlinear Schrödinger-KdV equation, J. Partial Differ. Equ., 12, 11-25 (1999) · Zbl 0931.35158
[15] Corcho, AJ; Linares, F., Well-posedness for the Schrödinger-Korteweg-de Vries system, Trans. Am. Math. Soc., 359, 4089-4106 (2007) · Zbl 1123.35063 · doi:10.1090/S0002-9947-07-04239-0
[16] Craig, W.; Guyenne, P.; Sulem, C., The surface signature of internal waves, J. Fluid Mech., 710, 277-303 (2012) · Zbl 1275.76059 · doi:10.1017/jfm.2012.364
[17] Craig, W.; Guyenne, P.; Sulem, C., Coupling between internal and surface waves, Nat. Hazards, 57, 617-642 (2011) · doi:10.1007/s11069-010-9535-4
[18] Davey, A.; Stewartson, K., On three-dimensional packets of surface waves, Proc. R. Soc. A, 338, 1613, 101-110 (1974) · Zbl 0282.76008
[19] Deconinck, B.; Nguyen, NV; Segal, BL, The interaction of long and short waves in dispersive media, J. Phys. A: Math. Theor., 49, 415501 (2016) · Zbl 1349.76030 · doi:10.1088/1751-8113/49/41/415501
[20] Dias, JP; Figueira, M.; Oliveira, F., Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system, Nonlinear Anal., 73, 2686-2698 (2010) · Zbl 1194.35405 · doi:10.1016/j.na.2010.06.049
[21] Djordjevic, VD; Redekopp, LG, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79, 4, 703-714 (1977) · Zbl 0351.76016 · doi:10.1017/S0022112077000408
[22] Funakoshi, M.; Oikawa, M., The resonant interactions between a long internal gravity wave and a surface gravity wave packet, J. Phys. Soc. Jpn., 52, 1982-1995 (1983) · doi:10.1143/JPSJ.52.1982
[23] Gandzha, IS; Sedletsky, YV; Dutykh, DS, High-order nonlinear Schrödinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions, Ukr. J. Phys., 59, 12, 1201-1214 (2014) · doi:10.15407/ujpe59.12.1201
[24] Gardner, CS; Greene, JM; Kruskal, MD; Miura, RM, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1061.35520 · doi:10.1103/PhysRevLett.19.1095
[25] Hasegawa, A.; Tappert, F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I, Anormalous dispersion, Appl. Phys. Lett., 23, 142 (1973) · doi:10.1063/1.1654836
[26] Hasegawa, A.; Tappert, F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers II, Normal dispersion, Appl. Phys. Lett., 23, 171 (1973) · doi:10.1063/1.1654847
[27] Ikezi, H., Nishikawa, K., Hojo, H., Mima, K.: Coupled electron-plasma and ion-acoustic solitons excited by parametric instability. In: Proceedings of the 5th International Conference on Plasma Physics and Contr. Nucl. Fusion Res., Tokyo, 239-248 (1974)
[28] Kawahara, T., Nonlinear self-modulation of capillary-gravity waves on liquid layer, J. Phys. Soc. Jpn., 38, 1, 265-270 (1975) · doi:10.1143/JPSJ.38.265
[29] Kawahara, T.; Sugimoto, N.; Kakutani, T., Nonlinear interaction between short and long capillary-gravity waves, J. Phys. Soc. Jpn., 39, 5, 1379-1386 (1975) · doi:10.1143/JPSJ.39.1379
[30] Ma, Y-C; Redekopp, LG, Some solutions pertaining to the resonant interaction of long and short waves, Phys. Fluids, 22, 10, 1872-1876 (1979) · Zbl 0408.76009 · doi:10.1063/1.862493
[31] McGoldrick, LF, On Wilton’s ripples: a special case of resonant interactions, J. Fluid Mech., 42, 1, 193-200 (1970) · Zbl 0208.56403 · doi:10.1017/S0022112070001179
[32] MITCHELL, M.; Segev, M., Self-trapping of inconherent white light, Nature, 387, 880-882 (1997) · doi:10.1038/43136
[33] Nishikawa, K.; Hojo, H.; Mima, K.; Ikezi, H., Coupled nonlinear electron-plasma and ion-acoustic waves, Phys. Rev. Lett., 33, 148-151 (1974) · doi:10.1103/PhysRevLett.33.148
[34] Rüegg, CH; Cavadini, N.; Furrer, A.; Güdel, H-U; Krämer, K.; Mutka, H.; Wildes, A.; Habicht, K.; Vorderwisch, P., Bose-Einstein condensation of the triple states in the magnetic insulator \(TlCuCl_3\), Nature, 423, 62-65 (2003) · doi:10.1038/nature01617
[35] Shabat, A.; Zakharov, V., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34, 62-69 (1972)
[36] Zakharov, VE, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 4, 190-194 (1968)
[37] Zakharov, VE, Collapse of Langmuir waves, Sov. Phys. Jetp., 35, 908-914 (1972)
[38] Wilton, JR, On ripples, Philos. Magn., 29, 6, 688-700 (1915) · JFM 45.1090.02 · doi:10.1080/14786440508635350
[39] Zakharov, VE, Collapse of Langmuir waves, Sov. Phys. JETP., 35, 908-914 (1972)
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.