Solution of a nonlinear Schrödinger equation in the form of two-phase
Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 1, pp. 89-103 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We construct a family of two-gap solutions of the focusing nonlinear Schrödinger equation and derive a condition under which the solutions behave as the so-called freak waves located at the nodes of a two-dimensional lattice. We also study how the lattice parameters depend on the parameters of the spectral curve.
Mots-clés : rogue wave
Keywords: freak wave, nonlinear Schrödinger equation, theta function, reduction, covering.
@article{TMF_2012_173_1_a4,
     author = {A. O. Smirnov},
     title = {Solution of a~nonlinear {Schr\"odinger} equation in the~form of two-phase},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {89--103},
     year = {2012},
     volume = {173},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_173_1_a4/}
}
TY  - JOUR
AU  - A. O. Smirnov
TI  - Solution of a nonlinear Schrödinger equation in the form of two-phase
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2012
SP  - 89
EP  - 103
VL  - 173
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2012_173_1_a4/
LA  - ru
ID  - TMF_2012_173_1_a4
ER  - 
%0 Journal Article
%A A. O. Smirnov
%T Solution of a nonlinear Schrödinger equation in the form of two-phase
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2012
%P 89-103
%V 173
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2012_173_1_a4/
%G ru
%F TMF_2012_173_1_a4
A. O. Smirnov. Solution of a nonlinear Schrödinger equation in the form of two-phase. Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 1, pp. 89-103. http://geodesic.mathdoc.fr/item/TMF_2012_173_1_a4/

[1] N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A, 373:6 (2009), 675–678 | DOI | Zbl

[2] N. Akhmediev, E. Pelinovsky, Eur. Phys. J. Spec. Top., 185:1 (2010), 1–4 | DOI

[3] A. I. Dyachenko, V. E. Zakharov, Pisma v ZhETF, 88:5 (2008), 356–359 | DOI

[4] V. E. Zakharov, R. V. Shamin, Pisma v ZhETF, 91:2 (2010), 68–71 | DOI

[5] N. N. Akhmediev, A. Ankevich, Solitony, Fizmatlit, M., 2003

[6] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, J. M. Dudley, Nature Physics, 6:10 (2010), 790–795 | DOI

[7] P. Dubard, P. Gaillard, C. Klein, V. B. Matveev, Eur. Phys. J. Spec. Top., 185:1 (2010), 247–258 | DOI

[8] P. Dubard, V. B. Matveev, Nat. Hazards Earth Syst. Sci., 11 (2011), 667–672 | DOI

[9] A. Ankiewicz, D. J. Kedzora, N. Akhmediev, Phys. Lett. A, 375:28–29 (2011), 2782–2785 | DOI | Zbl

[10] D. J. Kedzora, A. Ankiewicz, N. Akhmediev, Phys. Rev. E, 84:5 (2011), 056611, 7 pp. | DOI

[11] B. Guo, L. Ling, Q. P. Liu, Phys. Rev. E, 85 (2012), 026607, 9 pp., arXiv: 1108.2867 | DOI

[12] A. R. Osborne, M. Onorato, M. Serio, Phys. Lett. A, 275:5–6 (2000), 386–393 | DOI | MR | Zbl

[13] A. Calini, C. M. Schober, Phys. Lett. A, 298:5–6 (2002), 335–349 | DOI | MR | Zbl

[14] C. M. Schober, Eur. J. Mech. B Fluids, 25:5 (2006), 602–620 | DOI | MR | Zbl

[15] K. B. Dysthe, Proc. Roy. Soc. London. Ser. A, 369:1736 (1979), 105–114 | DOI | Zbl

[16] K. Trulsen, K. B. Dysthe, Wave Motion, 24:3 (1996), 281–289 | DOI | MR | Zbl

[17] K. Trulsen, I. Kliakhandler, K. B. Dysthe, M. G. Velarde, Phys. Fluids, 12:10 (2000), 2433–2437 | DOI | MR

[18] A. Saini, V. M. Vyas, S. N. Pandey, T. S. Raju, P. K. Panigrahi, Traveling wave solutions to nonlinear Schröedinger equation with self-steepening and self-frequency shift, arXiv: 0911.2788

[19] A. L. Islas, C. M. Schober, Phys. Fluids, 17:3 (2005), 031701, 4 pp. | DOI | MR | Zbl

[20] S. P. Novikov, Funkts. analiz i ego pril., 8:3 (1974), 54–66 | DOI | MR | Zbl

[21] P. D. Lax, “Periodic solutions of the KdV equations”, Nonlinear Wave Motion (Potsdam, NY, 1972), Lectures in Applied Mathematics, 15, ed. A. C. Newell, AMS, Providence, RI, 1974, 85–96 | MR

[22] B. A. Dubrovin, S. P. Novikov, Dokl. AN SSSR, 219:3 (1974), 19–22 | MR

[23] A. R. Its, V. B. Matveev, TMF, 23:1 (1975), 51–68 | DOI | MR

[24] B. A. Dubrovin, V. B. Matveev, S. P. Novikov, UMN, 31:1(187) (1976), 55–136 | DOI | MR | Zbl

[25] I. M. Krichever, UMN, 32:6(198) (1977), 183–208 | DOI | MR | Zbl

[26] B. A. Dubrovin, UMN, 36:2(218) (1981), 11–80 | DOI | MR | Zbl

[27] E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, V. B. Matveev, Algebro-geometrical Approach to Nonlinear Evolution Equations, Springer Series in Nonlinear Dynamics, Springer, Berlin, 1994 | Zbl

[28] C. Klein, O. Richter, Ernst Equation and Riemann Surfaces. Analitical and Numerical Methods, Lecture Notes in Physics, 685, Springer, Berlin, 2005 | MR | Zbl

[29] K. Klein, V. B. Matveev, A. O. Smirnov, TMF, 152:2 (2007), 304–320 | DOI | MR | Zbl

[30] A. I. Bobenko, C. Klein (eds.), Computational Approach to Riemann Surfaces, Lecture Notes in Mathematics, 2013, Springer, Berlin, 2011 | DOI | MR | Zbl

[31] C. Kalla, C. Klein, On the numerical evaluation of algebro-geometric solutions to integrable equations, arXiv: 1107.2108 | MR

[32] C. Kalla, C. Klein, New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation, arXiv: 1109.5301 | MR

[33] E. G. Amosenok, A. O. Smirnov, Lett. Math. Phys., 96:1–3 (2011), 157–168 | DOI | MR | Zbl

[34] A. O. Smirnov, G. M. Golovachev, E. G. Amosenok, Nelineinaya dinam., 7:2 (2011), 239–256

[35] D. W. McLaughlin, C. M. Schober, Physica D, 57:3–4 (1992), 447–465 | DOI | MR | Zbl

[36] M. J. Ablowitz, C. M. Schober, B. M. Herbst, Phys. Rev. Lett., 71:17 (1993), 2683–2686 | DOI

[37] A. Calini, N. M. Ercolani, D. W. McLaughlin, C. M. Schober, Physica D, 89:3–4 (1996), 227–260 | DOI | MR | Zbl

[38] A. R. Its, Vestnik LGU. Cer. Matem. Mekhan. Astronom., 7:2 (1976), 39–46 | MR | Zbl

[39] B. A. Dubrovin, I. M. Krichever, S. P. Novikov, Dokl. AN SSSR, 229:1 (1976), 15–19 | MR | Zbl

[40] J. D. Fay, Theta-functions on Riemann Surfaces, Lecture Notes in Mathematics, 352, Springer, Berlin, 1973 | DOI | MR | Zbl

[41] H. F. Baker, Abel's Theorem and the Allied Theory Including the Theory of Theta Functions, Cambridge Univ. Press, Cambridge, 1897 | MR | Zbl

[42] A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903 | MR | Zbl

[43] A. O. Smirnov, Matem. sb., 133(175):3(7) (1987), 382–391 | DOI | MR | Zbl

[44] N. I. Akhiezer, Elementy teorii ellipticheskikh funktsii, Nauka, M., 1970 | MR | Zbl

[45] A. O. Smirnov, Matem. zametki, 58:1 (1995), 86–97 | DOI | MR | Zbl

[46] A. O. Smirnov, Matem. sb., 185:8 (1994), 103–114 | DOI | MR | Zbl

[47] A. O. Smirnov, TMF, 107:2 (1996), 188–200 | DOI | MR | Zbl

[48] A. O. Smirnov, Matem. sb., 188:1 (1997), 109–128 | DOI | MR | Zbl

[49] A. O. Smirnov, TMF, 78:1 (1989), 11–21 | DOI | MR