Genuine nonlinearity and its connection to the modified Korteweg-de Vries equation in phase dynamics. (English) Zbl 1479.35748


35Q53 KdV equations (Korteweg-de Vries equations)
35C20 Asymptotic expansions of solutions to PDEs
35L65 Hyperbolic conservation laws
78A60 Lasers, masers, optical bistability, nonlinear optics
35Q60 PDEs in connection with optics and electromagnetic theory
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B55 Internal waves for incompressible inviscid fluids
76B70 Stratification effects in inviscid fluids
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI arXiv


[1] Ablowitz, M. J.; Cole, J. T.; Rumanov, I., Whitham equations and phase shifts for the Korteweg-de Vries equation, Proc. R. Soc. A, 476, 20200300 (2020) · Zbl 1472.35330
[2] Ablowitz, M. J.; Hammack, J.; Henderson, D.; Schober, C. M., Modulated periodic Stokes waves in deep water, Phys. Rev. Lett., 84, 887 (2000) · Zbl 0986.76008
[3] Agrawal, G. P., Nonlinear Fibre Optics (1989), New York: Academic, New York
[4] Akylas, T. R., Three-dimensional long water-wave phenomena, Annu. Rev. Fluid Mech., 26, 191-210 (1994) · Zbl 0802.76007
[5] Ankiewicz, A.; Akhmediev, N., Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions, Nonlinear Dyn., 91, 1931-1938 (2018)
[6] Barros, R.; Choi, W.; Milewski, P. A., Strongly nonlinear effects on internal solitary waves in three-layer flows, J. Fluid Mech., 883, A16 (2020) · Zbl 1430.76084
[7] Brandt, A.; Shipley, K. R., Laboratory experiments on mass transport by large amplitude mode-2 internal solitary waves, Phys. Fluids, 26 (2014)
[8] Benzoni-Gavage, S.; Noble, P.; Rodrigues, L. M., Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids, J. Nonlinear Sci., 24, 711-768 (2014) · Zbl 1308.35021
[9] Bridges, T. J., Symmetry, Phase Modulation, and Nonlinear Waves (2017), Cambridge: Cambridge University Press, Cambridge · Zbl 1383.76002
[10] Bridges, T. J.; Kostianko, A.; Zelik, S., Validity of the hyperbolic Whitham modulation equations in Sobolev spaces, J. Differ. Equ., 274, 971-995 (2021) · Zbl 1455.35233
[11] Bridges, T. J.; Kostianko, A.; Schneider, G., A proof of validity for multiphase Whitham modulation theory, Proc. R. Soc. A, 476, 20200203 (2020) · Zbl 1472.35332
[12] Bridges, T. J.; Ratliff, D. J., On the elliptic-hyperbolic transition in Whitham modulation theory, SIAM J. Appl. Math., 77, 1989-2011 (2017) · Zbl 1375.74047
[13] Bridges, T. J.; Ratliff, D. J., Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory, J. Nonlin. Sci., 31, 1-45 (2021) · Zbl 1476.37086
[14] Bronski, J. C.; Hur, V. M.; Johnson, M. A., Modulational instability in equations of KdV type, New Approaches to Nonlinear Waves, 83-133 (2016), Berlin: Springer, Berlin
[15] Carles, R.; Danchin, R.; Saut, J-C, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25, 2843 (2012) · Zbl 1251.35142
[16] Carr, M.; Davies, P. A.; Hoebers, R. P., Experiments on the structure and stability of mode-2 internal solitary-like waves propagating on an offset pycnocline, Phys. Fluids, 27 (2015)
[17] Carr, M.; Stastna, M.; Davies, P. A.; van de Wal, K. J., Shoaling mode-2 internal solitary-like waves, J. Fluid Mech., 879, 604-632 (2019) · Zbl 1430.76085
[18] Choi, W.; Camassa, R., Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396, 1-36 (1999) · Zbl 0973.76019
[19] Conforti, M.; Baronio, F.; Trillo, S., Resonant radiation shed by dispersive shock waves, Phys. Rev. A, 89 (2014)
[20] Conforti, M.; Trillo, S., Dispersive wave emission from wave breaking, Opt. Lett., 38, 3815-3818 (2013)
[21] Conforti, M.; Trillo, S.; Mussot, A.; Kudlinski, A., Parametric excitation of multiple resonant radiations from localized wavepackets, Sci. Rep., 5, 9433 (2015)
[22] Deepwell, D.; Stastna, M.; Carr, M.; Davies, P. A., Wave generation through the interaction of a mode-2 internal solitary wave and a broad, isolated ridge, Phys. Rev. Fluids, 4 (2019)
[23] Djordjevic, V. D.; Redekopp, L. G., The fission and disintegration of internal solitary waves moving over two-dimensional topography, J. Phys. Oceanogr., 8, 1016-1024 (1978)
[24] Doelman, A.; Sandstede, B.; Scheel, A.; Schneider, G., The Dynamics of Modulated Wave Trains (2009), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1179.35005
[25] El, G.; Hoefer, M. A., Dispersive shock waves and modulation theory, Physica D, 333, 11-65 (2016) · Zbl 1415.76001
[26] El, G.; Hoefer, M. A.; Shearer, M., Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws, SIAM Rev., 59, 3-61 (2017) · Zbl 1364.35307
[27] Esler, J. G.; Pearce, J. D., Dispersive dam-break and lock-exchange flows in a two-layer fluid, J. Fluid Mech., 667, 555-585 (2011) · Zbl 1225.76068
[28] Grava, T.; Pierce, V. U.; Tian, F. R., Initial value problem of the Whitham equations for the Camassa-Holm equation, Physica D, 238, 55 (2009) · Zbl 1161.35487
[29] Grimshaw, R. H J., Nonlinear aspects of long shelf waves, Geophys. Astrophys. Fluid Dyn., 8, 3-16 (1977) · Zbl 0353.76073
[30] Grimshaw, R. H J.; Ostrovsky, L. A.; Shrira, V. I.; Stepanyants, Y. A., Long nonlinear surface and internal gravity waves in a rotating ocean, Surv. Geophys., 19, 289-338 (1998)
[31] Grimshaw, R. H J.; Pelinovsky, E.; Talipova, T., The modified Korteweg-de Vries equation in the theory of large-amplitude internal waves, Nonlinear Proc. Geophys., 4, 237-250 (1997)
[32] Gurevich, A. V.; Krylov, L.; El, G. A., Quasilongitudinal nonlinear dispersing MHD waves, Zh. Eksp. Teor. Fiz., 102, 1524-1539 (1992)
[33] Ivanov, S. K., Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate, Phys. Rev. E, 96 (2017)
[34] Johnson, M. A.; Perkins, W. R., Modulational instability of viscous fluid conduit periodic waves, SIAM J. Math. Anal., 52, 277-305 (2020) · Zbl 1436.35280
[35] Kakutani, T.; Ono, H., Weak nonlinear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Japan, 26, 1305-1318 (1969)
[36] Kalisch, H.; Nguyen, N. T., On the stability of internal waves, J. Phys. A Math. Theor., 43 (2010) · Zbl 1420.76003
[37] Kamchatnov, A. M., Undular bore theory for the Gardner equation, Phys. Rev. E, 86 (2012)
[38] Kamchatnov, A. M., Evolution of initial discontinuities in the DNLS equation theory, J. Phys. Commun., 2 (2018)
[39] Koop, C. G.; Butler, G., An investigation of internal solitary waves in a two fluid system, J. Fluid Mech., 112, 225-251 (1981) · Zbl 0479.76036
[40] Konno, K.; Ichikawa, Y. H., A modified Korteweg de Vries equation for ion acoustic waves, J. Phys. Soc. Japan, 37, 1631-1636 (1974)
[41] Kuramoto, Y., Phase dynamics of weakly unstable periodic structures, Prog. Theor. Phys., 71, 1182-1196 (1984) · Zbl 1046.76502
[42] Lamb, K. G., Conjugate flows for a three-layer fluid, Phys. Fluids, 12, 2169-2185 (2000) · Zbl 1184.76308
[43] Lax, P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences (1973), Philadelphia, PA: SIAM, Philadelphia, PA · Zbl 0268.35062
[44] Leblond, H.; Grelu, P.; Mihalache, D., Models for supercontinuum generation beyond the slowly-varying-envelope approximation, Phys. Rev. A, 90 (2014)
[45] Maiden, M.; Hoefer, M. A., Modulations of viscous fluid conduit periodic waves, Proc. R. Soc. A, 472, 20160533 (2016) · Zbl 1371.76066
[46] Marchant, T. R., Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation, Wave Motion, 45, 540 (2008) · Zbl 1231.35208
[47] McLean, J. W., Instabilities of finite-amplitude water waves, J. Fluid Mech., 114, 315-330 (1982) · Zbl 0483.76027
[48] Miura, R., Korteweg-de Vries equation and generalizations: I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9, 1202 (1968) · Zbl 0283.35018
[49] Malaguti, S.; Conforti, M.; Trillo, S., Dispersive radiation induced by shock waves in passive resonators, Opt. Lett., 39, 5626-5629 (2014)
[50] Nakamura, Y.; Tsukabayashi, I., Observation of modified Korteweg-de Vries solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett., 52, 2356 (1984)
[51] Olbers, D.; Willebrand, J.; Eden, C., Ocean Dynamics (2012), Berlin: Springer, Berlin · Zbl 1296.86001
[52] Pierce, V. U.; Tian, F-R, Self-similar solutions of the non-strictly hyperbolic Whitham equations, Commun. Math. Sci., 4, 799 (2006) · Zbl 1133.35086
[53] Pomeau, Y.; Manneville, P., Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74, 189-197 (1980)
[54] Ratliff, D. J., Phase dynamics of periodic wavetrains leading to the 5th order KP equation, Physica D, 353, 11-19 (2017) · Zbl 1378.35272
[55] Ratliff, D. J., Conservation laws, modulation and the emergence of universal forms, PhD Thesis (2017)
[56] Ratliff, D. J., The modulation of multiple phases leading to the modified Korteweg de Vries equation, Chaos, 28 (2018) · Zbl 1406.35334
[57] Ratliff, D. J., Dispersive dynamics in the characteristic moving frame, Proc. R. Soc. A, 475, 20180784 (2019) · Zbl 1427.35242
[58] Schamel, H., A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons, J. Plasma Phys., 9, 377-387 (1973)
[59] Sprenger, P.; Hoefer, M. A., Discontinuous shock solutions of the Whitham modulation equations as zero dispersion limits of traveling waves, Nonlinearity, 33, 3268 (2020) · Zbl 1440.35212
[60] Stuart, J. T.; DiPrima, R. C., The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. R. Soc. A, 362, 27-41 (1978)
[61] Triki, H.; Leblond, H.; Mihalache, D., Derivation of a modified Korteweg-de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian, Opt. Commun., 28, 3179-3186 (2012)
[62] Trulsen, K.; Kliakhandler, I.; Dysthe, K. B.; Velarde, M. G., On weakly nonlinear modulation of waves on deep water, Phys. Fluids, 12, 2432-2437 (2000) · Zbl 1184.76558
[63] Whitham, G. B., Nonlinear dispersion of water waves, J. Fluid Mech., 27, 399-412 (1967) · Zbl 0146.23702
[64] Whitham, G. B., Linear and Nonlinear Waves, vol 42 (2011), New York: Wiley, New York · Zbl 0373.76001
[65] Willebrand, J., Energy transport in a nonlinear and inhomogeneous random gravity wave field, J. Fluid Mech., 70, 113-126 (1975) · Zbl 0317.76009
[66] Zhang, D. J.; Zhao, S. L.; Sun, Y. Y.; Zhou, J., Solutions to the modified Korteweg-de Vries equation, Rev. Math. Phys., 26, 1430006 (2014) · Zbl 1341.37049
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