Geodesic
Stabilization Mechanism for Two-Dimensional Solitons in Nonlinear Parametric Resonance
Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 2, pp. 226-233 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a simple model system supporting stable solitons in two dimensions. The system is the parametrically driven damped nonlinear Schrodinger equation, and the soliton stabilizes for sufficiently strong damping. We elucidate the stabilization mechanism by reducing the partial differential equation to a finite-dimensional dynamical system and conclude that the negative feedback loop occurs via enslaving the soliton phase, locked to the driver, to its amplitude and width.
Keywords: Faraday resonance, driven damped nonlinear Schrodinger equation, stability.
Mots-clés : oscillons 
@article{TMF_2005_144_2_a1,
     author = {N. V. Alekseeva},
     title = {Stabilization {Mechanism} for {Two-Dimensional} {Solitons} in {Nonlinear} {Parametric} {Resonance}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {226--233},
     year = {2005},
     volume = {144},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2005_144_2_a1/}
}
TY  - JOUR
AU  - N. V. Alekseeva
TI  - Stabilization Mechanism for Two-Dimensional Solitons in Nonlinear Parametric Resonance
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2005
SP  - 226
EP  - 233
VL  - 144
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2005_144_2_a1/
LA  - ru
ID  - TMF_2005_144_2_a1
ER  - 
%0 Journal Article
%A N. V. Alekseeva
%T Stabilization Mechanism for Two-Dimensional Solitons in Nonlinear Parametric Resonance
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2005
%P 226-233
%V 144
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2005_144_2_a1/
%G ru
%F TMF_2005_144_2_a1
N. V. Alekseeva. Stabilization Mechanism for Two-Dimensional Solitons in Nonlinear Parametric Resonance. Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 2, pp. 226-233. http://geodesic.mathdoc.fr/item/TMF_2005_144_2_a1/

[1] H. W. Müller, R. Friedrich, D. Papathanassiou, “Theoretical and experimental investigation of the Faraday instability”, Evolution of Spontaneous Structures in Dissipative Continuous Systems, Lecture Notes in Physics, 55, eds. F. Busse and S. C. Müller, Springer, N.Y., 1998, 231 | MR

[2] P. B. Umbanhowar, F. Melo, H. L. Swinney, Nature, 382 (1996), 793 | DOI

[3] O. Lioubashevski, H. Arbell, J. Fineberg, Phys. Rev. Lett., 76 (1996), 3959 ; H. Arbell, J. Fineberg, Phys. Rev. Lett., 85 (2000), 756 | DOI | DOI

[4] D. Astruc, S. Fauve, “Parametrically Amplified 2-Dimensional Solitary Waves”, IUTAM Symposium on Free Surface Flows, Proc. UIIAM Symp. on Free Surface Flows (Birmingham, UK, July 10–14, 2000), Fluid Mechanics and Its Applications, 62, eds. A. C. King and Y. D. Shikhmurzaev, Kluwer, Boston, 2001, 39 | DOI | MR | Zbl

[5] O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, J. Fineberg, Phys. Rev. Lett., 83 (1999), 3190 | DOI

[6] L. S. Tsimring, I. S. Aranson, Phys. Rev. Lett., 79 (1997), 213 ; H. Sakaguchi, H. R. Brand, Europhys. Lett., 38 (1997), 341 ; Physica D, 117 (1998), 95 ; C. Crawford, H. Riecke, Physica D, 129 (1999), 83 | DOI | DOI | DOI | Zbl | DOI

[7] E. Cerda, F. Melo, S. Rica, Phys. Rev. Lett., 79 (1997), 4570 ; S. C. Venkataramani, E. Ott, Phys. Rev. Lett., 80 (1998), 3495 ; S.-O. Jeong, T.-W. Ko, H.-T. Moon, Physica D, 164 (2002), 71 | DOI | DOI | DOI | Zbl

[8] D. Rothman, Phys. Rev. E, 57 (1998), 1239 | DOI

[9] J. Eggers, H. Riecke, Phys. Rev. E, 59 (1999), 4476 | DOI

[10] I. V. Barashenkov, N. V. Alexeeva, E. V. Zemlyanaya, Phys. Rev. Lett., 89 (2002), 104101 | DOI

[11] S. V. Kiyashko, L. N. Korzinov, M. I. Rabinovich, L. S. Tsimring, Phys. Rev. E, 54 (1996), 5037 ; W. Zhang, J. Viñals, Phys. Rev. Lett., 74 (1995), 690 | DOI | DOI

[12] V. J. Sánchez-Morcillo, I. Pérez-Arjona, F. Silva, G. J. de Valcárel, E. Roldán, Opt. Lett., 25 (2000), 957 ; I. Pérez-Arjona, E. Roldán, G. J. de Valcárcel, Squezzed quantum cavity solitons, E-print quant-ph/0405148 | DOI

[13] K. Rypdal, J. J. Rasmussen, K. Thomsen, Physica D, 16 (1985), 339 | DOI | MR | Zbl

[14] I. V. Barashenkov, M. M. Bogdan, V. I. Korobov, Europhys. Lett., 15 (1991), 113 | DOI

[15] E. A. Kuznetsov, S. K. Turitsyn, Phys. Lett. A, 112 (1985), 273 ; V. M. Malkin, E. G. Shapiro, Physica D, 53 (1991), 25 | DOI | DOI | MR | Zbl

[16] V. E. Zakharov, E. A. Kuznetsov, ZhETF, 91 (1986), 1310