Elliptic solitons and “freak waves”

Matveev, V.; Smirnov, A.

doi:10.1090/spmj/1713

Elliptic solitons and “freak waves”
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by V. B. Matveev and A. O. Smirnov
Translated by: V. B. Matveev

St. Petersburg Math. J. 33 (2022), 523-551

DOI: https://doi.org/10.1090/spmj/1713

Published electronically: May 5, 2022

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Abstract:

It is shown that elliptic solutions to the AKNS hierarchy equations can be obtained by exploring spectral curves that correspond to elliptic solutions of the KdV hierarchy. This also allows one to get the quasirational and trigonometric solutions for AKNS hierarchy equations as a limit case of the elliptic solutions mentioned above.

References

E. D. Belokolos, A. I. Bobenko, V. Z. Enol′skii, A. R. Its, and V. B. Matveev, Algebro-geometrical approach to nonlinear evolution equations, Springer Ser. Nonlinear Dynamics, Springer, Berlin, 1994.
K. Kleĭn, V. B. Matveev, and A. O. Smirnov, The cylindrical Kadomtsev-Petviashvili equation: old and new results, Teoret. Mat. Fiz. 152 (2007), no. 2, 304–320 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 152 (2007), no. 2, 1132–1145. MR 2429282, DOI 10.1007/s11232-007-0097-x
R. Fedele, S. De Nicola, D. Jovanović, D. Grecu, and A. Visinescu, On the mapping connecting the cylindrical nonlinear von Neumann equation with the standard von Neumann equation, J. Plasma Phys. 76 (2010), 645–653.
K. R. Khusnutdinova, C. Klein, V. B. Matveev, and A. O. Smirnov, On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation, Chaos 23 (2013), no. 1, 013126, 13. MR 3389617, DOI 10.1063/1.4792268
V. B. Matveev and A. O. Smirnov, Solutions of the Ablowitz-Kaup-Newell-Segur hierarchy equations of the “rogue wave” type: a unified approach, Teoret. Mat. Fiz. 186 (2016), no. 2, 191–220 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 186 (2016), no. 2, 156–182. MR 3462749, DOI 10.4213/tmf8958
Francesco Calogero and Antonio Degasperis, Spectral transform and solitons. Vol. I, Lecture Notes in Computer Science, vol. 144, North-Holland Publishing Co., Amsterdam-New York, 1982. Tools to solve and investigate nonlinear evolution equations. MR 680040
Vladimir B. Matveev and Aleksandr O. Smirnov, AKNS and NLS hierarchies, MRW solutions, $P_{n}$ breathers, and beyond, J. Math. Phys. 59 (2018), no. 9, 091419, 42. MR 3858688, DOI 10.1063/1.5049949
A. O. Smirnov, Solution of a nonlinear Schrödinger equation in the form of two-phase freak waves, Theoret. and Math. Phys. 173 (2012), no. 1, 1403–1416. Translation of Teoret. Mat. Fiz. 173 (2012), no. 1, 89–103. MR 3171538, DOI 10.1007/s11232-012-0122-6
A. O. Smirnov, Periodic two-phase “rogue waves”, Math. Notes 94 (2013), no. 5-6, 897–907. Translation of Mat. Zametki 94 (2013), no. 6, 871–883. MR 3227031, DOI 10.1134/S0001434613110266
A. O. Smirnov and G. M. Golovachev, Constructed in the elliptic functions three-phase solutions for the nonlinear Schrödinger equation, Rus. J. Nonlin. Dyn. 9 (2013), no. 3, 389–407.
Aleksandr O. Smirnov, Sergei G. Matveenko, Sergei K. Semenov, and Elena G. Semenova, Three-phase freak waves, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 032, 14. MR 3338006, DOI 10.3842/SIGMA.2015.032
E. D. Belokolos and V. Z. Ènol′skiĭ, Verdier’s elliptic solitons and the Weierstrass reduction theory, Funktsional. Anal. i Prilozhen. 23 (1989), no. 1, 57–58 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 1, 46–47. MR 998428, DOI 10.1007/BF01078572
E. D. Belokolos and V. Z. Ènol′skiĭ, Isospectral deformations of elliptic potentials, Uspekhi Mat. Nauk 44 (1989), no. 5(269), 155–156 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 5, 191–193. MR 1040275, DOI 10.1070/RM1989v044n05ABEH002212
V. Z. Ènol′skiĭ and N. A. Kostov, On the geometry of elliptic solitons, Acta Appl. Math. 36 (1994), no. 1-2, 57–86. MR 1303856, DOI 10.1007/BF01001543
E. D. Belokolos and V. Z. Ènol′skiĭ, Reduction of theta functions and elliptic finite-gap potentials, Acta Appl. Math. 36 (1994), no. 1-2, 87–117. MR 1303857, DOI 10.1007/BF01001544
Emma Previato, Victor Enolski (1945–2019), Notices Amer. Math. Soc. 67 (2020), no. 11, 1755–1767. MR 4201918, DOI 10.1090/noti
A. O. Smirnov, Finite-gap elliptic solutions of the KdV equation, Acta Appl. Math. 36 (1994), no. 1-2, 125–166. MR 1303859, DOI 10.1007/BF01001546
Alexander O. Smirnov, Elliptic solitons and Heun’s equation, The Kowalevski property (Leeds, 2000) CRM Proc. Lecture Notes, vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp. 287–305. MR 1916788, DOI 10.1090/crmp/032/16
V. B. Matveev and A. O. Smirnov, Multiphase solutions of nonlocal symmetric reductions of equations of the AKNS hierarchy: general analysis and simplest examples, Teoret. Mat. Fiz. 204 (2020), no. 3, 383–395 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 204 (2020), no. 3, 1154–1165. MR 4153747, DOI 10.4213/tmf9901
Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. MR 450815, DOI 10.1002/sapm1974534249
M. Lakshmanan, K. Porsezian, and M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A 133 (1988), 483–488.
K. Porsezian, M. Daniel, and M. Lakshmanan, On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain, J. Math. Phys. 33 (1992), no. 5, 1807–1816. MR 1159002, DOI 10.1063/1.529658
M. Daniel, K. Porsezian, and M. Lakshmanan, On the integrable models of the higher order water wave equation, Phys. Lett. A 174 (1993), no. 3, 237–240. MR 1206888, DOI 10.1016/0375-9601(93)90765-R
Ryogo Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Mathematical Phys. 14 (1973), 805–809. MR 338587, DOI 10.1063/1.1666399
Chao-Qing Dai and Jie-Fang Zhang, New solitons for the Hirota equation and generalized higher-order nonlinear Schrödinger equation with variable coefficients, J. Phys. A 39 (2006), no. 4, 723–737. MR 2201534, DOI 10.1088/0305-4470/39/4/002
Adrian Ankiewicz, J. M. Soto-Crespo, and Nail Akhmediev, Rogue waves and rational solutions of the Hirota equation, Phys. Rev. E (3) 81 (2010), no. 4, 046602, 8. MR 2736231, DOI 10.1103/PhysRevE.81.046602
Linjing Li, Zhiwei Wu, Lihong Wang, and Jingsong He, High-order rogue waves for the Hirota equation, Ann. Physics 334 (2013), 198–211. MR 3063237, DOI 10.1016/j.aop.2013.04.004
J. S. He, C. Z. Li, and K. Porsezian, Rogue waves of the Hirota and the Maxwell–Bloch equations, Phys. Rev. E 87 (2013), 012913.
L. H. Wang, K. Porsezian, and J. S. He, Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation, Phys. Rev. E 87 (2013), 053202.
Adrian Ankiewicz and Nail Akhmediev, Higher-order integrable evolution equation and its soliton solutions, Phys. Lett. A 378 (2014), no. 4, 358–361. MR 3146431, DOI 10.1016/j.physleta.2013.11.031
Amdad Chowdury, Wieslaw Krolikowski, and N. Akhmediev, Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits, Phys. Rev. E 96 (2017), no. 4, 042209, 13. MR 3819432, DOI 10.1103/physreve.96.042209
A. Chowdury, D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions, Phys. Rev. E (3) 91 (2015), no. 2, 022919, 11. MR 3418612, DOI 10.1103/PhysRevE.91.022919
A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, Infinite hierarchy of nonlinear Schrödinger equations and their solutions, Phys. Rev. E 93 (2016), no. 1, 012206, 10. MR 3673204, DOI 10.1103/physreve.93.012206
A. Ankiewicz and N. Akhmediev, Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy, Phys. Rev. E 96 (2017), no. 1, 012219, 8. MR 3814079, DOI 10.1103/physreve.96.012219
A. O. Smirnov and V. B. Matveev, Some comments on continuous symmetries of AKNS hierarchy equations and their solutions, Preprint, arXiv:1509.1134, 2015.
D. J. Kedziora, A. Ankiewicz, A. Chowdury, and N. Akhmediev, Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients, Chaos 25 (2015), no. 10, 103114, 11. MR 3401584, DOI 10.1063/1.4931710
A. R. Its and V. P. Kotlyarov, On a class of solutions of the nonlinear Schrödinger equation, Dokl. Akad. Nauk Ukrain. SSR. Ser. A 11 (1976), 965–968, (Russian).
A. R. Its, Inversion of hyperelliptic integrals, and integration of nonlinear differential equations, Vestnik Leningrad. Univ. 7 Mat. Meh. Astronom. vyp. 2 (1976), 39–46, 162 (Russian, with English summary). MR 0609747
V. P. Kotlyarov, Periodic problem for the nonlinear Schrödinger equation, Preprint, arXiv: 1401.4445, 2014.
A. O. Smirnov, Elliptic solutions of the nonlinear Schrödinger equation and a modified Korteweg-de Vries equation, Mat. Sb. 185 (1994), no. 8, 103–114 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 82 (1995), no. 2, 461–470. MR 1302625, DOI 10.1070/SM1995v082n02ABEH003575
A. O. Smirnov, Solutions of the nonlinear Schrödinger equation that are elliptic in $t$, Teoret. Mat. Fiz. 107 (1996), no. 2, 188–200 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 107 (1996), no. 2, 568–578. MR 1406550, DOI 10.1007/BF02071370
A. O. Smirnov, E. G. Semenova, V. Zinger, and N. Zinger, On a periodic solution of the focusing nonlinear Schrödinger equation, Preprint, arXiv:1407.7974, 2014.
John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
W., Literaturberichte: Lehrbuch der Thetafunktionen, Monatsh. Math. Phys. 16 (1905), no. 1, A10–A11 (German). von Adolf Krazer. (Teubners Sammlung von Lehrbüchern auf dem Gebiete der mathematischen Wissenschaften. Band XII. Leipzig, B. G. Teubner 1903. [XXIV+563 gr. 8$^\textrm {o}$]). MR 1547425, DOI 10.1007/BF01693790
H. F. Baker, Abelian functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Abel’s theorem and the allied theory of theta functions; Reprint of the 1897 original; With a foreword by Igor Krichever. MR 1386644
David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651, DOI 10.1007/978-1-4899-2843-6
David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776, DOI 10.1007/978-0-8176-4578-6
B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
A. O. Smirnov, Elliptic solutions of the Korteweg-de Vries equation, Mat. Zametki 45 (1989), no. 6, 66–73, 111 (Russian); English transl., Math. Notes 45 (1989), no. 5-6, 476–481. MR 1019038, DOI 10.1007/BF01158237
A. O. Smirnov, Finite-gap solutions of the abelian Toda lattice of genus $4$ and $5$ in elliptic functions, Teoret. Mat. Fiz. 78 (1989), no. 1, 11–21 (Russian, with English summary); English transl., Theoret. and Math. Phys. 78 (1989), no. 1, 6–13. MR 987408, DOI 10.1007/BF01016911
N. I. Akhiezer, Elements of the theory of elliptic functions, Translations of Mathematical Monographs, vol. 79, American Mathematical Society, Providence, RI, 1990. Translated from the second Russian edition by H. H. McFaden. MR 1054205, DOI 10.1090/mmono/079
M. Abramowitz and I. A. Stegun (Eds), Handbook of mathematical functions with formulas, graphs and mathematical tables, Nat. Bureau Standards Appl. Math. Ser., vol. 55, Washington, D.C., 1972.