The finite-gap method and the periodic Cauchy problem for $(2+1)$-dimensional anomalous waves for the focusing Davey–Stewartson $2$ equation

P. G. Grinevich   ; P. M. Santini

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 6, pp. 1029-1059

Résumé

The focusing nonlinear Schrödinger equation is the simplest universal model describing the modulation instability of $(1+1)$-dimensional quasi monochromatic waves in weakly nonlinear media, and modulation instability is considered to be the main physical mechanism for the appearance of anomalous (rogue) waves in nature. By analogy with the recently developed analytic theory of periodic anomalous waves of the focusing nonlinear Schrödinger equation, in this paper we extend these results to a $(2+1)$-dimensional context, concentrating on the focusing Davey–Stewartson $2$ equation, an integrable $(2+1)$-dimensional generalization of the focusing nonlinear Schrödinger equation. More precisely, we use the finite gap theory to solve, to the leading order, the doubly periodic Cauchy problem for the focusing Davey–Stewartson $2$ equation, for small initial perturbations of the unstable background solution, which we call the doubly periodic Cauchy problem for anomalous waves. As in the case of the nonlinear Schrödinger equation, we show that, to the leading order, the solution of this Cauchy problem is expressed in terms of elementary functions of the initial data. Bibliography: 86 titles.

Détail

Keywords: Davey–Stewartson equation, rogue (anomalous) waves in multidimensional problems, doubly periodic Cauchy problem, finite-gap integration, asymptotic solutions.

Bibliographie

[1] M. J. Ablowitz, G. Biondini, and S. Blair, “Nonlinear Schrödinger equations with mean terms in nonresonant multidimensional quadratic materials”, Phys. Rev. E (3), 63:4 (2001), 046605 | DOI

[2] M. J. Ablowitz and R. Haberman, “Nonlinear evolution equations – two and three dimensions”, Phys. Rev. Lett., 35:18 (1975), 1185–1188 | DOI | MR

[3] M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM Stud. Appl. Math., 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1981, x+425 pp. | DOI | MR | Zbl

[4] N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves”, J. Opt., 15:6 (2013), 060201, 9 pp. | DOI

[5] N. N. Akhmediev, V. M. Eleonskiĭ, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions”, Zh. Èksper. Teoret. Fiz., 89:5 (1985), 1542–1551 ; English transl. in Soviet Phys. JETP, 62:5 (1985), 894–899

[6] D. Anker and N. C. Freeman, “On the soliton solutions of the Davey–Stewartson equation for long waves”, Proc. Roy. Soc. London Ser. A, 360:1703 (1978), 529–540 | DOI | MR | Zbl

[7] H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions”, Phys. Rev. Lett., 107:25 (2011), 255005 | DOI

[8] F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, M. Onorato, and S. Wabnitz, “Vector rogue waves and baseband modulation instability in the defocusing regime”, Phys. Rev. Lett., 113:3 (2014), 034101, 5 pp. | DOI

[9] E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro-geometric approach to nonlinear integrable equations, Springer Ser. Nonlinear Dynam., Springer-Verlag, Berlin, 1994, xii+337 pp. | Zbl

[10] T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. Part I. Theory”, J. Fluid Mech., 27 (1967), 417–430 | DOI | Zbl

[11] D. J. Benney and G. J. Roskes, “Wave instabilities”, Stud. Appl. Math., 48:4 (1969), 377–385 | DOI | Zbl

[12] V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids”, Pis'ma Zh. Èksper. Teoret. Fiz., 3:12 (1966), 471–476; English transl. in JETP Lett., 3:12 (1966), 307–310

[13] Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves”, Phys. Rev. A, 80:3 (2009), 033610 | DOI

[14] M. Boiti, J. J.-P. Leon, L. Martina, and F. Pempinelli, “Scattering of localized solitons in the plane”, Phys. Lett. A, 132:8-9 (1988), 432–439 | DOI | MR

[15] A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank”, Phys. Rev. Lett., 106:20 (2011), 204502 | DOI

[16] I. V. Cherednik, “Reality conditions in ‘finite-zone integration’”, Dokl. Akad. Nauk SSSR, 252:5 (1980), 1104–1108 ; English transl. in Soviet Phys. Dokl., 25:6 (1980), 450–452 | MR | Zbl

[17] F. Coppini, P. G. Grinevich, and P. M. Santini, “Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics”, Phys. Rev. E, 101:3 (2020), 032204, 8 pp. | DOI | MR

[18] F. Coppini, P. G. Grinevich, and P. M. Santini, “Periodic rogue waves and perturbation theory”, Encyclopedia of complexity and systems science, Springer, Berlin–Heidelberg, 2022, 1–22, Publ. online | DOI

[19] F. Coppini and P. M. Santini, “The Fermi–Pasta–Ulam–Tsingou recurrence of periodic anomalous waves in the complex Ginzburg–Landau and in the Lugiato–Lefever equations”, Phys. Rev. E, 102:6 (2020), 062207, 11 pp. | DOI | MR

[20] A. Davey and K. Stewartson, “On three-dimensional packets of surface waves”, Proc. Roy. Soc. London Ser. A, 338:1613 (1974), 101–110 | DOI | MR | Zbl

[21] G. Dematteis, T. Grafke, M. Onorato, and E. Vanden-Eijnden, “Experimental evidence of hydrodynamic instantons: the universal route to rogue waves”, Phys. Rev. X, 9 (2019), 041057, 12 pp. | DOI

[22] B. A. Dubrovin, “Theta functions and non-linear equations”, Uspekhi Mat. Nauk, 36:2(218) (1981), 11–80 ; English transl. in Russian Math. Surveys, 36:2 (1981), 11–92 | MR | Zbl | DOI

[23] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “The Schrödinger equation in a periodic field and Riemann surfaces”, Dokl. Akad. Nauk SSSR, 229:1 (1976), 15–18 ; English transl. in Soviet Math. Dokl., 17 (1977), 947–951 | MR | Zbl

[24] K. B. Dysthe and K. Trulsen, “Note on breather type solutions of the NLS as models for freak-waves”, Phys. Scr., T82:1 (1999), 48–52 | DOI

[25] G. A. El, “Soliton gas in integrable dispersive hydrodynamics”, J. Stat. Mech. Theory Exp., 2021, no. 11, 114001, 69 pp. | DOI | MR | Zbl

[26] G. A. El and A. Tobvis, “Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation”, Phys. Rev. E, 101:5 (2020), 052207, 21 pp. | DOI | MR

[27] J. D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Math., 352, Springer-Verlag, Berlin–New York, 1973, iv+137 pp. | DOI | MR | Zbl

[28] F. Fedele, J. Brennan, S. Ponce de León, J. Dudley, and F. Dias, “Real world ocean rogue waves explained without the modulational instability”, Sci. Rep., 6 (2016), 27715 | DOI

[29] A. S. Fokas and P. M. Santini, “Coherent structures in multidimensions”, Phys. Rev. Lett., 63:13 (1989), 1329–1333 | DOI | MR

[30] A. S. Fokas and P. M. Santini, “Dromions and a boundary value problem for the Davey–Stewartson 1 equation”, Phys. D, 44:1-2 (1990), 99–130 | DOI | MR | Zbl

[31] A. Gelash, D. Agafontsev, V. Zakharov, G. El, S. Randoux, and P. Suret, “Bound state soliton gas dynamics underlying the spontaneous modulational instability”, Phys. Rev. Lett., 123:23 (2019), 234102 | DOI

[32] P. G. Grinevich, A. E. Mironov, and S. P. Novikov, “Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-$1/2$ particles”, Teor. Mat. Fiz., 164:3 (2010), 333–353 ; “Erratum”, 166:2 (2011), 320 ; English transl. in Theoret. and Math. Phys., 164:3 (2010), 1110–1127 ; 166:2 (2011), 278 | DOI | Zbl | DOI | DOI | DOI

[33] P. G. Grinevich and P. M. Santini, “The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1”, Nonlinearity, 31:11 (2018), 5258–5308 | DOI | MR

[34] P. G. Grinevich and P. M. Santini, “The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes”, Phys. Lett. A, 382:14 (2018), 973–979 | DOI | MR | Zbl

[35] P. G. Grinevich and P. M. Santini, “The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes”, Uspekhi Mat. Nauk, 74:2(446) (2019), 27–80 ; English transl. in Russian Math. Surveys, 74:2 (2019), 211–263 | DOI | MR | Zbl | DOI

[36] S. Haver, Freak wave event at Draupner jacket January 1 1995, Tech. Rep. PTT-KU-MA, Statoil, Oslo, 2003

[37] K. L. Henderson, D. H. Peregrine, and J. W. Dold, “Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation”, Wave Motion, 29:4 (1999), 341–361 | DOI | MR | Zbl

[38] G. Huang, L. Deng, and C. Hang, “Davey–Stewartson description of two-dimensional nonlinear excitations in Bose–Einstein condensates”, Phys. Rev. E, 72:3 (2005), 036621 | DOI

[39] A. R. Its, A. V. Rybin, and M. A. Sall, “Exact integration of nonlinear Schrödinger equation”, Teor. Mat. Fiz., 74:1 (1988), 29–45 ; English transl. in Theoret. and Math. Phys., 74:1 (1988), 20–32 | MR | Zbl | DOI

[40] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics”, Nature Phys., 6:10 (2010), 790–795 | DOI

[41] B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Diaz, and J. M. Dudley, “Observation of Kuznetsov–Ma soliton dynamics in optical fibre”, Sci. Rep., 2 (2012), 463, 5 pp. | DOI

[42] Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications”, Phys. Rep., 298:2-3 (1998), 81–197 | DOI

[43] C. Klein and K. Roidot, “Numerical study of the semiclassical limit of the Davey–Stewartson II equations”, Nonlinearity, 27:9 (2014), 2177–2214 | DOI | MR

[44] C. Klein and J.-C. Saut, “IST versus PDE: a comparative study”, Hamiltonian partial differential equations and applications, Fields Inst. Commun., 75, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, 383–449 | DOI | MR | Zbl

[45] C. Klein and N. Stoilov, “Numerical study of blow-up mechanisms for Davey–Stewartson II systems”, Stud. Appl. Math., 141:1 (2018), 89–112 | DOI | MR | Zbl

[46] B. G. Konopelchenko, “Induced surfaces and their integrable dynamics”, Stud. Appl. Math., 96:1 (1996), 9–51 | DOI | MR | Zbl

[47] B. G. Konopelchenko, “Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy”, Ann. Global Anal. Geom., 18:1 (2000), 61–74 | DOI | MR | Zbl

[48] I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications”, Uspekhi Mat. Nauk, 44:2(266) (1989), 121–184 ; English transl. in Russian Math. Surveys, 44:2 (1989), 145–225 | MR | Zbl | DOI

[49] I. M. Krichever, Perturbation theory in periodic problems for two-dimensional integrable systems, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., 9, Part 2, Harwood Acad. Publ., Reading, UK, 1992, 103 pp. | Zbl

[50] E. A. Kuznetsov, “Solitons in a parametrically unstable plasma”, Dokl. Akad. Nauk SSSR, 236:3 (1977), 575–577 ; English transl. in Soviet Phys. Dokl., 22 (1977), 507–508

[51] C. Liu, R. E. C. van der Wel, N. Rotenberg, L. Kuipers, T. F. Krauss, A. Di Falco, and A. Fratalocchi, “Triggering extreme events at the nanoscale in photonic seas”, Nature Phys., 11:4 (2015), 358–363 | DOI

[52] C. Liu, C. Wang, Z. Dai, and J. Liu, “New rational homoclinic and rogue waves for Davey–Stewartson equation”, Abstr. Appl. Anal., 2014 (2014), 572863, 8 pp. | DOI | MR | Zbl

[53] Y. Liu, C. Qian, D. Mihalache, and J. He, “Rogue waves and hybrid solutions of the Davey–Stewartson I equation”, Nonlinear Dynam., 95:1 (2019), 839–857 | DOI | Zbl

[54] R. M. Matuev and I. A. Taimanov, “The Moutard transformation of two-dimensional Dirac operators and the conformal geometry of surfaces in four-dimensional space”, Mat. Zametki, 100:6 (2016), 868–880 ; English transl. in Math. Notes, 100 (2016), 835–846 | DOI | MR | Zbl | DOI

[55] W. M. Moslem, R. Sabry, S. K. El-Labany, and P. K. Shukla, “Dust-acoustic rogue waves in a nonextensive plasma”, Phys. Rev. E, 84:6 (2011), 066402 | DOI

[56] D. Mumford, Tata lectures on theta, v. I, Progr. Math., 28, Birkhäuser Boston, Inc., Boston, MA, 1983, xiii+235 pp. ; v. II, 43, 1984, xiv+272 pp. | DOI | MR | Zbl | DOI | MR | Zbl

[57] A. C. Newell and J. V. Moloney, Nonlinear optics, Adv. Top. Interdiscip. Math. Sci., Addison-Wesley Publishing Co., Redwood City, CA, 1992, xii+436 pp. | MR | Zbl

[58] K. Nishinari, K. Abe, and J. Satsuma, “A new type of soliton behavior of the Davey–Stewartson equations in a plasma system”, Teor. Mat. Fiz., 99:3 (1994), 487–498 ; English transl. in Theoret. and Math. Phys., 99:3 (1994), 745–753 | MR | Zbl | DOI

[59] Y. Ohta and Jianke Yang, “Rogue waves in the Davey–Stewartson I equation”, Phys. Rev. E, 86:3 (2012), 036604 | DOI

[60] Y. Ohta and Jianke Yang, “Dynamics of rogue waves in the Davey–Stewartson II equation”, J. Phys. A, 46:10 (2013), 105202, 19 pp. | DOI | MR | Zbl

[61] M. Onorato, T. Waseda, A. Toffoli, L. Cavaleri, O. Gramstad, P. A. E. M. Janssen, T. Kinoshita, J. Monbaliu, N. Mori, A. R. Osborne, M. Serio, C. T. Stansberg, H. Tamura, and K. Trulsen, “Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events”, Phys. Rev. Lett., 102:11 (2009), 114502 | DOI

[62] A. R. Osborne, M. Onorato, and M. Serio, “The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains”, Phys. Lett. A, 275:5-6 (2000), 386–393 | DOI | MR | Zbl

[63] T. Ozawa, “Exact blow-up solutions to the Cauchy problem for the Davey–Stewartson systems”, Proc. Roy. Soc. London Ser. A, 436:1897 (1992), 345–349 | DOI | MR | Zbl

[64] F. Pedit and U. Pinkall, “Quaternionic analysis on Riemann surfaces and differential geometry”, Proceedings of the international congress of mathematicians, Vol. II (Berlin 1998), Doc. Math., Extra Vol. II (1998), 389–400 | MR | Zbl

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

[66] D. Pierangeli, M. Flammini, L. Zhang, G. Marcucci, A. J. Agranat, P. G. Grinevich, P. M. Santini, C. Conti, and E. DelRe, “Observation of Fermi–Pasta–Ulam–Tsingou recurrence and its exact dynamics”, Phys. Rev. X, 8:4 (2018), 041017, 9 pp. | DOI

[67] D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves”, Nature, 450:7172 (2007), 1054–1057 | DOI

[68] I. A. Taimanov, “Modified Novikov–Veselov equation and differential geometry of surfaces”, Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, 179, Adv. Math. Sci., 33, Amer. Math. Soc., Providence, RI, 1997, 133–151 | DOI | MR | Zbl

[69] I. A. Taimanov, “The global Weierstrass representation and its spectrum”, Uspekhi Mat. Nauk, 52:6(318) (1997), 187–188 ; English transl. in Russian Math. Surveys, 52:6 (1997), 1330–1332 | DOI | MR | Zbl | DOI

[70] I. A. Taimanov, “The Weierstrass representation of closed surfaces in $\mathbb R^3$”, Funktsional. Anal. Prilozhen., 32:4 (1998), 49–62 ; English transl. in Funct. Anal. Appl., 32:4 (1998), 258–267 | DOI | MR | Zbl | DOI

[71] I. A. Taimanov, “On two-dimensional finite-gap potential Schrödinger and Dirac operators with singular spectral curves”, Sibirsk. Mat. Zh., 44:4 (2003), 870–882 ; English transl. in Siberian Math. J., 44:4 (2003), 686–694 | MR | Zbl | DOI

[72] I. A. Taimanov, “Surfaces in the four-space and the Davey–Stewartson equations”, J. Geom. Phys., 56:8 (2006), 1235–1256 | DOI | MR | Zbl

[73] I. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces”, Uspekhi Mat. Nauk, 61:1(367) (2006), 85–164 ; English transl. in Russian Math. Surveys, 61:1 (2006), 79–159 | DOI | MR | Zbl | DOI

[74] I. A. Taimanov, “Singular spectral curves in finite-gap integration”, Uspekhi Mat. Nauk, 66:1(397) (2011), 111–150 ; English transl. in Russian Math. Surveys, 66:1 (2011), 107–144 | DOI | MR | Zbl | DOI

[75] I. A. Taimanov, “The Moutard transformation of two-dimensional Dirac operators and Möbius geometry”, Mat. Zametki, 97:1 (2015), 129–141 ; English transl. in Math. Notes, 97:1 (2015), 124–135 | DOI | MR | Zbl | DOI

[76] I. A. Taimanov, “Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces”, Teor. Mat. Fiz., 182:2 (2015), 213–222 ; English transl. in Theoret. and Math. Phys., 182:2 (2015), 173–181 | DOI | MR | Zbl | DOI

[77] I. A. Taimanov, “The Moutard transformation for the Davey–Stewartson II equation and its geometrical meaning”, Mat. Zametki, 110:5 (2021), 751–765 ; English transl. in Math. Notes, 110:5 (2021), 754–766 | DOI | MR | Zbl | DOI

[78] I. A. Taimanov and S. P. Tsar{e}v, “Blowing up solutions of the Novikov–Veselov equation”, Dokl. Ross. Akad. Nauk, 420:6 (2008), 744–745 ; English transl. in Dokl. Math., 77:3 (2008), 467–468 | MR | Zbl | DOI

[79] A. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations”, Dokl. Akad. Nauk SSSR, 279:1 (1984), 20–24 ; English transl. in Soviet Math. Dokl., 30:3 (1984), 588–591 | MR | Zbl

[80] A. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional Schrödinger operators. Potential operators”, Dokl. Akad. Nauk SSSR, 279:4 (1984), 784–788 ; English transl. in Soviet Math. Dokl., 30:3 (1984), 705–708 | MR | Zbl

[81] L. Wen, L. Li, Z. D. Li, S. W. Song, X. F. Zhang, and W. M. Liu, “Matter rogue wave in Bose–Einstein condensates with attractive atomic interaction”, Eur. Phys. J. D, 64 (2011), 473–478 | DOI

[82] V. E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, Prikl. Mekh. Tekhn. Fiz., 1968, no. 2, 86–94; English transl. in J. Appl. Mech. Tech. Phys., 9:2 (1968), 190–194 | DOI

[83] V. E. Zakharov and A. A. Gelash, “Nonlinear stage of modulation instability”, Phys. Rev. Lett., 111:5 (2013), 054101 | DOI

[84] V. E. Zakharov and A. M. Rubenchik, “Nonlinear interaction of high-frequency and low-frequency waves”, Prikl. Mekh. Tekhn. Fiz., 1972, no. 5, 84–98; English transl. in J. Appl. Mech. Tech. Phys., 13:5 (1972), 669–681 | DOI

[85] V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Zh. Èksper. Teoret. Fiz., 61:1 (1971), 118–134 ; English transl. in Soviet Phys. JETP, 34:1 (1972), 62–69 | MR

[86] V. E. Zakharov and A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I”, Funktsional. Anal. Prilozhen., 8:3 (1974), 43–53 ; English transl. in Funct. Anal. Appl., 8:3 (1974), 226–235 | MR | Zbl | DOI