Geodesic
A novel variational approach to pulsating solitons in the cubic-quintic Ginzburg–Landau equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 152 (2007) no. 2, pp. 339-355
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Comprehensive numerical simulations of pulse solutions of the cubic-quintic Ginzburg–Landau equation (CGLE) reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary wave solutions, i.e., pulsating, creeping, snake, erupting, and chaotic solitons that are not stationary in time. They are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcation sequences of these pulses as the CGLE parameters are varied. We address the issues of central interest in this area, i.e., the conditions for the occurrence of the five categories of dissipative solitons and also the dependence of both their shape and their stability on the various CGLE parameters, i.e., the nonlinearity, dispersion, linear and nonlinear gain, loss, and spectral filtering. Our predictions for the variation of the soliton amplitudes, widths, and periods with the CGLE parameters agree with the simulation results. We here present detailed results for the pulsating solitary waves. Their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with the simulation results. We will address snakes and chaotic solitons in subsequent papers. This overall approach fails to address only the dissipative solitons in one class, i.e., the exploding or erupting solitons.
Mots-clés : variational formalism
Keywords: complex Ginzburg–Landau equation, pulsating soliton. 
@article{TMF_2007_152_2_a9,
     author = {S. C. Mancas and S. R. Choudhury},
     title = {A~novel variational approach to pulsating solitons in the~cubic-quintic {Ginzburg{\textendash}Landau} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {339--355},
     year = {2007},
     volume = {152},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2007_152_2_a9/}
}
TY  - JOUR
AU  - S. C. Mancas
AU  - S. R. Choudhury
TI  - A novel variational approach to pulsating solitons in the cubic-quintic Ginzburg–Landau equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2007
SP  - 339
EP  - 355
VL  - 152
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2007_152_2_a9/
LA  - ru
ID  - TMF_2007_152_2_a9
ER  - 
%0 Journal Article
%A S. C. Mancas
%A S. R. Choudhury
%T A novel variational approach to pulsating solitons in the cubic-quintic Ginzburg–Landau equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2007
%P 339-355
%V 152
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2007_152_2_a9/
%G ru
%F TMF_2007_152_2_a9
S. C. Mancas; S. R. Choudhury. A novel variational approach to pulsating solitons in the cubic-quintic Ginzburg–Landau equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 152 (2007) no. 2, pp. 339-355. http://geodesic.mathdoc.fr/item/TMF_2007_152_2_a9/

[1] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, 1982 | MR | Zbl

[2] P. G. Drazin, W. H. Reid, Hydrodynamic Stability, Cambridge Monogr. Mech. Appl. Math., Cambridge Univ. Press, Cambridge, 1981 | MR | Zbl

[3] I. S. Aranson, L. Kramer, Rev. Mod. Phys., 74 (2002), 99 | DOI | MR | Zbl

[4] C. Bowman, A. C. Newell, Rev. Mod. Phys., 70 (1998), 289 | DOI

[5] L. Brusch, A. Torcini, M. Bär, Physica D, 174 (2003), 152 | DOI | MR | Zbl

[6] L. Brusch, A. Torcini, M. van Hecke, M. G. Zimmermann, M. Bär, Physica D, 160 (2001), 127 | DOI | MR | Zbl

[7] L. R. Keefe, Stud. Appl. Math., 73 (1985), 91 | DOI | MR | Zbl

[8] M. J. Landman, Stud. Appl. Math., 76 (1987), 187 | DOI | MR | Zbl

[9] P. K. Newton, L. Sirovich, Quart. Appl. Math., 44 (1986), 49 | DOI | MR | Zbl

[10] P. K. Newton, L. Sirovich, Quart. Appl. Math., 44 (1986), 367 | DOI | MR | Zbl

[11] W. van Saarloos, P. C. Hohenberg, Physica D, 56 (1992), 303 | DOI | MR | Zbl

[12] M. van Hecke, C. Storm, W. van Saarloos, Physica D, 134 (1999), 1 | DOI | MR | Zbl

[13] R. Alvarez, M. van Hecke, W. van Saarloos, Phys. Rev. E, 56 (1997), R1306 | DOI

[14] P. Holmes, Physica D, 23 (1986), 84 | DOI | MR | Zbl

[15] A. Doelman, J. Nonlinear Sci., 3 (1993), 225 | DOI | MR | Zbl

[16] A. Doelman, Physica D, 53 (1991), 249 | DOI | MR | Zbl

[17] J. Duan, P. Holmes, Proc. Edinburgh Math. Soc. (2), 38 (1995), 77 | DOI | MR | Zbl

[18] A. Doelman, Physica D, 40 (1989), 156 | DOI | MR | Zbl

[19] J. M. Soto-Crespo, N. Akhmediev, G. Town, Opt. Commun., 199 (2001), 283 | DOI

[20] N. Akhmediev, J. Soto-Crespo, G. Town, Phys. Rev. E, 63 (2001), 056602 | DOI

[21] D. Artigas, L. Torner, N. Akhmediev, Opt. Commun., 143 (1997), 322 | DOI

[22] J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl., 1974, no. 55, 284 | DOI | MR

[23] P. G. Drazin, R. S. Johnson, Solitons: an Introduction, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[24] J. D. Murray, Mathematical Biology, Biomathematics, 19, Springer, Berlin, 1989 | MR | Zbl

[25] N. J. Balmforth, Annual Rev. Fluid Mech., 27 (1995), 335 | DOI | MR

[26] N. Akhmediev, “Pulsating solitons in dissipative systems”, Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory, IV IMACS Int. Conf. (Athens, GA, 2005)

[27] D. J. Kaup, B. A. Malomed, Physica D, 87 (1995), 155 | DOI | MR | Zbl

[28] N. Akhmediev, A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations”, Dissipative Solitons, Lecture Notes in Phys., 661, eds. N. Akhmediev, A. Ankiewicz, Springer, Berlin, 2005, 1 | DOI | MR | Zbl

[29] L. A. Takhtadzhyan, L. D. Faddeev, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | MR | Zbl | Zbl

[30] V. Skarka, “Generation and dynamics of dissipative spatial solitons”, Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory, IV IMACS Int. Conf. (Athens, GA, 2005)

[31] A. H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics. Analytical, Computational, and Experimental Methods, Wiley Ser. Nonlinear Sci., Wiley, New York, 1995 | MR | Zbl

[32] M. Holodniok, M. Kubíček, Computation of period doubling bifurcation points in ordinary differential equations, Report TUM-M8406, Inst. Math. Informatik, Technischen Universität, München, 1984 | Zbl