Geodesic
Two-phase periodic solutions to the AKNS hierarchy equations
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 25, Tome 473 (2018), pp. 205-227 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we investigate the genus $2$ algebro-geometric solutions of the AKNS hierarchy equations, strictly periodic with respect to the space variable $x$. In general position these solutions, expressed by means of two-dimensional Riemann theta functions are not strictly periodic in $x$. We show that $x$ periodic solutions can be obtained by appropriate choice of the hyperelliptic spectral curves, having a structure of covering over elliptic curve. For odd number members of AKNS hierarchy these solutions might be made periodic also with respect to the corresponding time variables of the AKNS hierarchy, imposing further restrictions on the structure of the spectral curve pointed out in the paper. The related solutions are especially interesting from the point of view of potential applications to study the signals propagation in nonlinear optical fibers. 
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V. B. Matveev; A. O. Smirnov. Two-phase periodic solutions to the AKNS hierarchy equations. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 25, Tome 473 (2018), pp. 205-227. http://geodesic.mathdoc.fr/item/ZNSL_2018_473_a12/

[1] A. O. Smirnov, V. B. Matveev, Some comments on continuous symmetries of AKNS hierarchy equations and their solutions, 2015, arXiv: 1509.1134 | MR

[2] V. B. Matveev, A. O. Smirnov, “Resheniya tipa “voln-ubiits” uravnenii ierarkhii Ablovitsa–Kaupa–Nyuella–Sigura: edinyi podkhod”, TMF, 186:2 (2016), 191–220 | DOI | Zbl

[3] V. B. Matveev, A. O. Smirnov, “AKNS and NLS hierarchies, MRW solutions, $P_n$ breathers, and beyond”, J. Math. Physics, 59 (2018), 091419–091462 | DOI | MR

[4] V. E. Zakharov, L. D. Faddeev, “Korteweg-de Vries equation: A completely integrable Hamiltonian system”, Functional Analysis and Its Applications, 5:4 (1971), 280–287 | DOI | MR

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

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

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

[8] R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation”, J. Math. Phys., 14 (1973), 805 | DOI | MR | Zbl

[9] C. Q. Dai, J. F. Zhang, “New solitons for the Hirota equation and generalized higher-order nonlinear Schrödinger equation with variable coefficients”, J. Phys. A, 39 (2006), 723–737 | DOI | MR | Zbl

[10] A. Ankiewicz, J. M. Soto-Crespo, N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation”, Phys. Rev. E, 81 (2010), 046602 | DOI | MR

[11] L. Li, Zh. Wu, L. Wang, J. He, “High-order rogue waves for the Hirota equation”, Annals of Physics, 334 (2013), 198–211 | DOI | MR | Zbl

[12] J. S. He, Ch. Zh. Li, K. Porsezian, “Rogue waves of the Hirota and the Maxwell-Bloch equations”, Phys. Rev. E, 87:1 (2013), 012913 | DOI

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

[14] A. Ankiewicz, N. Akhmediev, “High-order integrable evolution equation and its soliton solutions”, Phys. Lett. A, 378 (2014), 358–361 | DOI | MR | Zbl

[15] A. Chowdury, W. Krolikowski, N. Akhmediev, “Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits”, Phys. Rev. E, 96:4 (2017), 042209 | DOI | MR

[16] A. Chowdury, D. J. Kedziora, A. Ankiewicz, N. Akhmediev, “Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions”, Phys. Rev. E, 91 (2015), 022919 | DOI | MR

[17] A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions”, Phys. Rev. E, 93:1 (2016), 012206 | DOI | MR

[18] A. Ankiewicz, N. Akhmediev, “Rogue wave-type solutions for the infinite integrable nonlinear Schrödinger hierarchy”, Phys. Rev. E, 96:1 (2017), 012219 | DOI | MR

[19] D. Kedziora, A. Ankiewicz, A. Chowdury, N. Akhmediev, “Integrable equations of the infinite nonlinear Schrödingier equation hierarchy with time variable coefficients”, Chaos, 25 (2015), 103114 | DOI | MR | Zbl

[20] A. R. Its, V. P. Kotlyarov, “Ob odnom klasse reshenii nelineinogo uravneniya Shredingera”, DAN USSR, Ser. A, 11 (1976), 965–968

[21] E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, V. B. Matveev, Algebro-geometrical approach to nonlinear evolution equations, Springer Ser. Nonlinear Dynamics, Springer, 1994 | MR

[22] F. Gesztesy, H. Holden, Soliton equation and their algebro-geometric solutions, v. 1, Cambridge Stud. in Adv. Math., 79, $(1+1)$-dimensional continuous models, Cambridge University Press, 2003 | MR

[23] A. O. Smirnov, “Ellipticheskie resheniya nelineinogo uravneniya Shredingera i modifitsirovannogo uravneniya Kortevega-de Friza”, Matem. Sbornik, 185:8 (1994), 103–114 | Zbl

[24] A. O. Smirnov, “Ellipticheskie po $t$ resheniya nelineinogo uravneniya Shredingera”, TMF, 107:2 (1996), 188–200 | DOI | Zbl

[25] B. A. Dubrovin, “Teta-funktsii i nelineinye uravneniya”, UMN, 36:2 (1981), 11–80 | Zbl

[26] G. F. Beiker, Abelevy funktsii. Teorema Abelya i svyazannaya s nei teoriya teta-funktsii, MTsNMO, M., 2008

[27] G. Springer, Introduction to Riemann surfaces, Addison-Wesley, 1957 | MR | Zbl

[28] H. M. Farkas, I. Kra, Riemann surfaces, Springer, New York, 1980 | MR | Zbl

[29] D. Mamford, Lektsii o teta-funktsiyakh, M., Mir, 1988

[30] J. D. Fay, Theta-functions on {R}iemann surfaces, Lect. Notes Math., 352, Springer, 1973 | DOI | MR | Zbl

[31] A. O. Smirnov, “Matrichnyi analog teoremy Appelya i reduktsii mnogomernykh teta-funktsii Rimana”, Matem. Sbornik, 175:7 (1987), 382–391 | Zbl

[32] A. O. Smirnov, “Konechnozonnye resheniya abelevoi tsepochki Tody roda 4 i 5 v ellipticheskikh funktsiyakh”, TMF, 78:1 (1989), 11–21

[33] N. I. Akhiezer, Elementy teorii ellipticheskikh funktsii, Nauka, M., 1970

[34] A. O. Smirnov, V. B. Matveev, Yu. A. Gusman, N. V Landa, Spectral curves for the rogue waves, 2017, arXiv: 1712.09309 | MR

[35] A. O. Smirnov, “Reshenie nelineinogo uravneniya Shredingera v vide dvukhfaznykh strannykh voln”, TMF, 173:1 (2012), 89–103 | DOI | Zbl

[36] A. O. Smirnov, “Periodicheskie dvukhfaznye “volny-ubiitsy””, Matem. Zametki, 94:6 (2013), 871–883 | DOI | Zbl