Geodesic
Bifurcations of solitary waves
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (2008), pp. 529-550.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper provides a brief review of the recent results devoted to bifurcations of solitary waves. The main attention is paid to the universality of soliton behavior and stability of solitons while approaching supercritical bifurcations. Near the transition point from supercritical to subcritical bifurcations, the stability of two families of solitons is studied in the framework of the generalized nonlinear Schrodinger equation. It is shown that one-dimensional solitons corresponding to the family of supercritical bifurcations are stable in the Lyapunov sense. The solitons from the subcritical bifurcation branch are unstable. The development of this instability results in the collapse of solitons. Near the time of collapse, the pulse amplitude and its width exhibit a self-similar behavior with a small asymmetry in the pulse tails due to self-steepening. 
@article{JMAG_2008_4_a3,
     author = {E. A. Kuznetsov and D. S. Agafontsev and F. Dias},
     title = {Bifurcations of solitary waves},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {529--550},
     publisher = {mathdoc},
     volume = {4},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2008_4_a3/}
}
TY  - JOUR
AU  - E. A. Kuznetsov
AU  - D. S. Agafontsev
AU  - F. Dias
TI  - Bifurcations of solitary waves
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2008
SP  - 529
EP  - 550
VL  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JMAG_2008_4_a3/
LA  - en
ID  - JMAG_2008_4_a3
ER  - 
%0 Journal Article
%A E. A. Kuznetsov
%A D. S. Agafontsev
%A F. Dias
%T Bifurcations of solitary waves
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2008
%P 529-550
%V 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JMAG_2008_4_a3/
%G en
%F JMAG_2008_4_a3
E. A. Kuznetsov; D. S. Agafontsev; F. Dias. Bifurcations of solitary waves. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (2008), pp. 529-550. http://geodesic.mathdoc.fr/item/JMAG_2008_4_a3/

[1] Soliton Theory, Plenum Press, New York–London, 1984 | MR | Zbl

[2] A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985 | MR

[3] Nauka, Moscow, 1995, 521 pp. | MR | Zbl

[4] M. S. Longuet-Higgins, “Capillary-Gravity Waves of Solitary Type on Deep Water”, J. Fluid Mech., 200 (1989), 451–470 | DOI | MR | Zbl

[5] G. Iooss, K. Kirchgässner, “Bifurcation d'Ondes Solitaires en Présence d'une Faible Tension superficielle”, C.R. Acad. Sci. Paris, Sér. I, 311 (1990), 265–268 | MR | Zbl

[6] J.-M. Vanden-Broeck, F. Dias, “Gravity-Capillary Solitary Waves in Water of Infinite Depth and Related Free-Surface Flows”, J. Fluid Mech., 240 (1992), 549–557 | DOI | MR | Zbl

[7] F. Dias, G. Iooss, “Gravity-Capillary Solitary Waves with Damped Oscillations”, Physica D, 65 (1993), 399–423 | DOI | MR | Zbl

[8] M. S. Longuet-Higgins, “Capillary-Gravity Waves of Solitary Type and Envelope Solitons on Deep Water”, J. Fluid Mech., 252 (1993), 703–711 | DOI | MR | Zbl

[9] T. R. Akylas, “Envelope Solitons with Stationary Crests”, Phys. Fluids, 5 (1993), 789–791 | DOI | Zbl

[10] JETP, 86 (1998), 1035–1046 | DOI

[11] JETP, 89 (1999), 163 ; L. D. Landau, E. M. Lifshitz, Fluid mechanics, 3rd Engl. Ed., Pergamon Press, New York, 1986 | DOI

[12] 3rd Ed., Nauka, Moscow, 1986

[13] F. Dias, G. Iooss, “Capillary-Gravity Interfacial Waves in Deep Water”, Eur. J. Mech. B/Fluids, 15 (1996), 367–390 | MR

[14] G. Iooss, “Existence d'Orbites Homoclines à un Équilibre Elliptique, pour un Système Réversible”, C.R. Acad. Sci. Paris, 324 (1997), 933–997 | DOI | MR

[15] V. E. Zakharov, E. A. Kuznetsov, “Hamiltonian Formalism for Nonlinear Waves”, Usp. Fiz. Nauk, 167 (1997), 1137–1167 (Russian) | DOI

[16] J. Nycander, “Steady Vortices in Plasmas and Geophysical Flows”, Chaos, 4 (1994), 253 | DOI

[17] JETP, 83 (1996), 73 ; “Short Opt. Solitons in Fibers”, Chaos, 10 (2000), 551–558 | DOI

[18] Sov. Phys. JETP, 34 (1972), 62–69 | MR

[19] V. E. Zakharov, Handbook of Plasma Physics. 2. Basic Plasma Physics II, eds. A. Galeev, R. Sudan, North–Holland, Amsterdam, 1984

[20] N. G. Vakhitov, A. A. Kolokolov, “Stationary Solutions of the Wave Equation in Media with Saturated Nonlinearity”, Izv. VUZ. Radiofizika, 16 (1973), 1020–1028

[21] E. A. Kuznetsov, A. M. Rubenchik, V. E. Zakharov, “Soliton Stability in Plasmas and Fluids”, Phys. Rep., 142 (1986), 103 | DOI | MR

[22] E. A. Kuznetsov, S. K. Turitsyn, “Talanov Transformations for Selffocusing Problems and Instability of Waveguides”, Phys. Let., 112A (1985), 273 | DOI

[23] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Fizmatgiz, Moscow, 1961 (Russian)

[24] L. Nirenberg, “An Extended Interpolation Inequality”, Ann. Sci. Norm. Sup. Pisa, 20:4 (1966), 733–737 | MR | Zbl

[25] M. I. Weinstein, “Nonlinear Schrodinger Equations and Sharp Interpolation Estimates”, Comm. Math. Phys., 87 (1983), 567–576 | DOI | MR | Zbl

[26] E. A. Kuznetsov, “Wave Collapse in Plasmas and Fluids”, Chaos, 6 (1996), 381–390 | DOI | Zbl

[27] E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal, S. K. Turitsyn, “Sharper Criteria of the Wave Collapse”, Physica D, 87 (1995), 273–284 | DOI | MR | Zbl

[28] D. S. Agafontsev, F. Dias, E. A. Kuznetsov, “Bifurcations and Stability of Internal Solitary Waves”, Pis'ma v ZhETF, 83:5 (2006), 241 (Russian)

[29] D. S. Agafontsev, F. Dias, E. A. Kuznetsov, “Deep-Water Internal Solitary Waves Near Critical Density Ratio”, Physica D, 225 (2007), 153 | DOI | MR | Zbl

[30] D. S. Agafontsev, F. Dias, E. A. Kuznetsov, “Collapse of Solitary Waves near Transition from Supercritical to Subcritical Bifurcations”, Pis'ma v ZhETF, 87 (2008), 767–771 (Russian)

[31] G. B. Whitham, “Nonlinear Dispersive Waves”, Proc. Rod. Soc. A, 283 (1965), 238 | DOI | MR | Zbl

[32] D. S. Agafontsev, “Deep-Water Internal Solitary Waves near Critical Density Ratio”, Pis'ma v ZhETF, 87 (2008), 225 (Russian)

[33] Electrodynamics of Continuous Media, 2-nd ed., London–Pergamon, 1984 | MR

[34] V. I. Petviashvili, “On the Equation for Unusual Soliton”, Fizika Plasmy, 2 (1976), 469

[35] D. J. Kaup, A. C. Newell, “Stationary Solutions of the Wave Equation in the Medium with Saturated Nonlinearity”, J. Math. Phys., 19:4 (1978), 798–801 | DOI | MR | Zbl

[36] V. E. Zakharov, “Collapse of Langmure Waves”, Z. Eksp. Teor. Fiz.; Sov. Phys. JETP, 35 (1972), 908

[37] S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged Description of Wave Beams in Linear and Nonlinear Media (the Method of Moments)”, Izv. VUZ. Radiofizika, 14 (1971), 1353 (Russian); Radiophys. Quant. Electron., 14 (1974), 1062 | DOI

[38] V. E. Zakharov, “Collapse and Self-Focusing of Langmuir Waves”, Handbook of Plasma Physics, Basic Plasma Physics, 2, eds. A. A. Galeev, R. N. Sudan, Elsevier, North-Holland, 1984

[39] S. K. Turitsyn, “Nonstable Solitons and Sharp Criteria for Wave Collapse”, Phys. Rev. E, 47 (1993), R1316 | MR

[40] N. A. Zharova, A. G. Litvak, V. A. Mironov, “Self-Focusing of Wave Packets and Envelope Shock Formation in Nonlinear Dispersive Media”, Z. Eksp. Teor. Fiz., 130 (2006), 21

[41] A. A. Balakin, A. G. Litvak, V. A. Mironov, S. A. Skobelev, “Structural Features of the Self-Action Dynamics of Ultrashort Electromagnetic Pulses”, Zh. Eksp. Teor. Fiz., 131 (2007), 408

[42] Sov. Phys. JETP, 61 (1985), 228 | MR