Quasi-rational solutions of nonlinear Schr\"odinger equation
Russian journal of nonlinear dynamics, Tome 11 (2015) no. 2, pp. 219-240.

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The method for constructing quasi-rational solutions of the nonlinear Schrödinger equation, Kadomtsev – Petviashvili equation and some other integrable nonlinear equations is considered. Examples of range 2 and 3 solutions are given.
Mots-clés : rogue waves, KP equation, Darboux transformation.
Keywords: freak waves, nonlinear Schrödinger equation
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V. B. Matveev; P. Dubard; A. O. Smirnov. Quasi-rational solutions of nonlinear Schr\"odinger equation. Russian journal of nonlinear dynamics, Tome 11 (2015) no. 2, pp. 219-240. http://geodesic.mathdoc.fr/item/ND_2015_11_2_a2/

[1] Zakharov V. E., “Stabilnost periodicheskikh voln konechnoi amplitudy na poverkhnosti glubokoi zhidkosti”, Zhurn. prikl. mekh. i tekhn. fiz., 9:2 (1968), 86–94

[2] Chiao R., Garmire E., Townes C. H., “Self-trapping of optical beams”, Phys. Rev. Lett., 13:15 (1964), 479–482

[3] Kuznetsov E. A., “Solitony v parametricheski nestabilnoi plazme”, Dokl. AN SSSR, 236:1–3 (1977), 575 pp.

[4] Yan Zh., “Vector financial rogue waves”, Phys. Lett. A, 375:48 (2011), 4274–4279

[5] Akhmediev N. N., Ankevich A., Solitony: Nelineinye impulsy i puchki, Fizmatlit, Moskva, 2003, 304 pp.

[6] “Discussion debate: Rogue waves — towards a unifying concept?”, Eur. Phys. J. Special Topics, 185 (2010), Springer, Berlin, 266 pp.

[7] Belokolos E. D., Bobenko A. I., Enol'skii V. Z., Its A. R., Matveev V. B., Algebro-geometric approach to nonlinear integrable equations, Springer Ser. Nonlinear Dynamics, Springer, Berlin, 1994, 337 pp.

[8] Matveev V. B., Salle M. A., Darboux transformations and solitons, Springer Ser. Nonlinear Dynamics, Springer, Berlin, 1991, 120 pp.

[9] Matveev V. B., “Darboux transformation and explicit solutions of the Kadomtcev – Petviaschvily equation, depending on functional parameters”, Lett. Math. Phys., 3:3 (1979), 213–216

[10] Matveev V. B., “Darboux transformation and the explicit solutions of differential-difference and difference-difference evolution equation: 1”, Lett. Math. Phys., 3:3 (1979), 217–222

[11] Matveev V. B., “Some comments on the rational solutions of the Zakharov – Schabat equations”, Lett. Math. Phys., 3:6 (1979), 503–512

[12] Matveev V. B., Salle M. A., “Differential-difference evolution equation: 2. Darboux transformation for the Toda lattice”, Lett. Math. Phys., 3:5 (1979), 425–429

[13] Sall M. A., “Preobrazovanie Darbu dlya neabelevykh i nelokalnykh uravnenii tipa tsepochki Tody”, TMF, 53:2 (1982), 227–237

[14] Sall M. A., “$L-A$ pary s ratsionalnoi zavisimostyu ot spektralnykh parametrov. Preobrazovanie {D}arbu”, Zap. nauchn. semin. LOMI, 161 (1987), 72–75

[15] Matveev V. B., “Darboux transformations, covariance theorems and integrable systems”, L. D. Faddeev's Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 201, ed. M. Semenov-Tian-Shansky, AMS, Providence, R.I., 2000, 179–209

[16] Peregrine D. H., “Water waves, nonlinear Schrödinger equations and their solutions”, J. Austral. Math. Soc. Ser. B, 25:1 (1983), 16–43

[17] Akhmediev N., Ankiewicz A., Taki M., “Waves that appear from nowhere and disappear without a trace”, Phys. Lett. A, 373:6 (2009), 675–678

[18] Akhmediev N. N., Eleonskii V. M., Kulagin N. E., “Generatsiya periodicheskoi posledovatelnosti pikosekundnykh impulsov v opticheskom volokne. Tochnye resheniya”, ZhETF, 89:5 (1985), 1542–1551

[19] Eleonskii V. M., Krichever I. M., Kulagin N. E., “Ratsionalnye mnogosolitonnye resheniya nelineinogo uravneniya Shrëdingera”, Dokl. AN SSSR, 287:3 (1986), 606–610

[20] Akhmediev N., Ankiewicz A., Soto-Crespo J. M., “Rogue waves and rational solutions of the nonlinear Schrödinger equation”, Phys. Rev. E, 80:2 (2009), 026601, 9 pp.

[21] Dubard P., Multi-rogue solutions to the focusing NLS equation, PhD Thesis, 2010 https://tel.archives-ouvertes.fr/tel-00625446

[22] Dubard P., Matveev V. B., “Multi-rogue waves solutions: From the NLS equation to the KP-I equation”, Nonlinearity, 26:12 (2013), R93–R125

[23] Its A. R., Rybin A. V., Sall M. A., “K voprosu o tochnom integrirovanii nelineinogo uravneniya Shrëdingera”, TMF, 74:1 (1988), 29–45

[24] Dubard P., Matveev V. B., “Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation”, Nat. Hazards Earth Syst. Sci., 11 (2011), 667–672

[25] Akhmediev N. N, Eleonskii V. M., Kulagin N. E., “Tochnye resheniya pervogo poryadka nelineinogo uravneniya Shrëdingera”, TMF, 72:2 (1987), 183–196

[26] Gesztesy F., Holden H., Soliton equation and their algebro-geometric solutions Vol. 1: $(1+1)$-dimensional continuous models, Cambridge Stud. in Adv. Math., 79, Cambridge Univ. Press, Cambridge, 2003, 505 pp.

[27] Lakshmanan M., Porsezian K., Daniel M., “Effect of discreteness on the continuum limit of the Heisenberg spin chain”, Phys. Lett. A, 133:9 (1988), 483–488

[28] Porsezian K., Daniel M., Lakshmanan M., “On the integrability aspects of the one-dimensional classical continuum isotropic Heisenberg spin chain”, J. Math. Phys., 33:5 (1992), 1807–1816

[29] Daniel M., Porsezian K., Lakshmanan M., “On the integrable models of the higher order water wave equation”, Phys. Lett. A, 174:3 (1993), 237–240

[30] Wang L. H., Porsezian K., He J. S., “Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation”, Phys. Rev. E, 87:5 (2013), 053202, 10 pp.

[31] Ankiewicz A., Akhmediev N., “High-order integrable evolution equation and its soliton solutions”, Phys. Lett. A, 378:4 (2014), 358–361

[32] Hirota R., “Exact envelope-soliton solutions of a nonlinear wave equation”, J. Math. Phys., 14 (1973), 805–809

[33] Dai Ch.-Q., Zhang J.-F., “New solitons for the Hirota equation and generalized higher-order nonlinear {S}chrödinger equation with variable coefficients”, J. Phys. A, 39:4 (2006), 723–737

[34] Ankiewicz A., Soto-Crespo J. M., Akhmediev N., “Rogue waves and rational solutions of the Hirota equation”, Phys. Rev. E (3), 81:4 (2010), 046602, 8 pp.

[35] Dubard P., Gaillard P., Klein C., Matveev V. B., “On multi-rogue waves solutions of the focusing NLS equation and positon solutions of the KdV equation”, Discussion debate: Rogue waves — towards a unifying concept?, Eur. Phys. J. Special Topics, 185, eds. N. Akhmediev, E. Pelinovsky, Springer, Berlin, 2010, 247–258

[36] Smirnov A. O., “Ellipticheskii brizer nelineinogo uravneniya Shrëdingera”, Zap. nauchn. semin. POMI, 398 (2012), 209–222

[37] Smirnov A. O., “Finite-gap elliptic solutions of the KdV equation”, Acta Appl. Math., 36:1–2 (1994), 125–166

[38] Smirnov A. O., Golovachev G. M., “Trekhfaznye resheniya nelineinogo uravneniya Shrëdingera v ellipticheskikh funktsiyakh”, Nelineinaya dinamika, 9:3 (2013), 389–407

[39] Gaillard P., Gastineau M., “Eighteen parameter deformations of the Peregrine breather of order ten solutions of the NLS equation”, Int. J. Mod. Phys. C., 26:2 (2015), 1550016, 14 pp.