Geodesic
Envelope solitary waves and dark solitons at a water--ice interface
Informatics and Automation, Selected issues of mathematics and mechanics, Tome 289 (2015), pp. 163-177.

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The article is devoted to the study of some self-focusing and defocusing features of monochromatic waves in basins with horizontal bottom under an ice cover. The form and propagation of waves in such basins are described by the full 2D Euler equations. The ice cover is modeled by an elastic Kirchhoff–Love plate and is assumed to be of considerable thickness so that the inertia of the plate is taken into account in the formulation of the model. The Euler equations involve the additional pressure from the plate that is freely floating at the surface of the fluid. Obviously, the self-focusing is closely connected with the existence of so-called envelope solitary waves, for which the envelope speed (group speed) is equal to the speed of filling (phase speed). In the case of defocusing, solitary envelope waves are replaced by so-called dark solitons. The indicated families of solitary waves are parametrized by the wave propagation speed and bifurcate from the quiescent state. The dependence of the existence of envelope solitary waves and dark solitons on the basin's depth is investigated. 
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     title = {Envelope solitary waves and dark solitons at a water--ice interface},
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A. T. Il'ichev. Envelope solitary waves and dark solitons at a water--ice interface. Informatics and Automation, Selected issues of mathematics and mechanics, Tome 289 (2015), pp. 163-177. http://geodesic.mathdoc.fr/item/TRSPY_2015_289_a8/

[1] Ilichev A. T., Uedinennye volny v modelyakh gidromekhaniki, Fizmatlit, M., 2003

[2] Iooss G., Kirchgässner K., “Bifurcation d'ondes solitaires en présence d'une faible tension superficielle”, C. r. Acad. sci. Paris Sér. 1, 311:5 (1990), 265–268 | MR | Zbl

[3] Dias F., Iooss G., “Capillary-gravity solitary waves with damped oscillations”, Physica D, 65:4 (1993), 399–323 | DOI | MR

[4] Benjamin T. B., Feir J. E., “The disintegration of wave trains on deep water. I: Theory”, J. Fluid Mech., 27 (1967), 417–430 | DOI | Zbl

[5] Bridges T. J., Mielke A., “A proof of the Banjamin–Feir instability”, Arch. Ration. Mech. Anal., 133:2 (1995), 145–198 | DOI | MR | Zbl

[6] Kirchgässner K., “Wave solutions of reversible systems and applications”, J. Diff. Eqns., 45:1 (1982), 113–127 | DOI | MR

[7] Iooss G., Pérouème M. C., “Perturbed homoclinic solutions in reversible 1:1 resonance vector fields”, J. Diff. Eqns., 102:1 (1993), 62–88 | DOI | MR | Zbl

[8] Iooss G., Adelmeyer M., Topics in bifurcation theory and applications, 2nd ed., World Scientific, Singapore, 1998 | MR | Zbl

[9] Haragus M., Iooss G., Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems, Springer, London, 2011 | MR | Zbl

[10] Pliss V. A., “O printsipe svedeniya v teorii ustoichivosti dvizheniya”, DAN SSSR, 154:5 (1964), 1044–1046 | MR | Zbl

[11] Mielke A., “Reduction of quasilinear elliptic equations in cylindrical domains with applications”, Math. Methods Appl. Sci., 10:1 (1988), 51–66 | DOI | MR | Zbl

[12] Vanderbauwhede A., Iooss G., “Center manifold theory in infinite dimensions”, Dyn. Rep. Expo. Dyn. Syst. New Ser., 1 (1992), 125–163 | MR | Zbl

[13] Elphick C., Tirapegui E., Brachet M. E., Coullet P., Iooss G., “A simple global characterization for normal forms of singular vector fields”, Physica D, 29:1–2 (1987), 95–127 | DOI | MR | Zbl

[14] Müller A., Ettema R., “Dynamic response of an ice-breaker hull to ice breaking”, Proc. 7th IAHR Int. Symp. Ice (Hamburg, 1984), v. 2, Hamburg, 1984, 287–296

[15] Love A. E. H., A treatise on the mathematical theory of elasticity, Univ. Press, Cambridge, 1927 | Zbl

[16] Forbes L. K., “Surface waves of large amplitude beneath an elastic sheet. I: High-order series solution”, J. Fluid Mech., 169 (1986), 409–428 | DOI | MR | Zbl

[17] Iooss G., Kirchgässner K., “Water waves for small surface tension: An approach via normal form”, Proc. R. Soc. Edinb. A, 122 (1992), 267–299 | DOI | MR | Zbl

[18] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR

[19] Ilichev A. T., Tomashpolskii V. Ya., “Solitonopodobnye struktury na poverkhnosti zhidkosti pod ledyanym pokrovom”, TMF, 182:2 (2015), 277–293 | DOI

[20] Dias F., Iooss G., “Ondes solitaires “noires” à l'interface entre deux fluides en présence de tension superficielle”, C. r. Acad. sci. Paris Sér. 1, 319:1 (1994), 89–93 | MR | Zbl

[21] Dias F., Iooss G., “Capillary-gravity interfacial waves in infinite depth”, Eur. J. Mech. B: Fluids, 15:3 (1996), 367–393 | MR | Zbl