Riemann Surfaces Close to Degenerate Ones in the Theory of Rogue Waves

P. G. Grinevich

Informatics and Automation, Geometry, Topology, and Mathematical Physics, Tome 325 (2024), pp. 93-118

Résumé

At present the development of the theory of rogue waves (also known as anomalous waves or freak waves) is one of priority directions in mathematical physics. At the moment there is no unanimous consensus about the generation mechanism of these waves (and it cannot be ruled out that the main contribution in different systems is made by different mechanisms); however, the main candidate is considered to be modulation instability in nonlinear media. One of the research directions in the theory of rogue waves involves the use of integrable models, including the nonlinear Schrödinger (NLS) equation. The choice of the latter is motivated by the fact that in the classical studies of V. I. Bespalov–V. I. Talanov and V. E. Zakharov in the mid-1960s, the NLS equation was derived as a model for describing modulation instability in nonlinear optics and in the theory of deep water waves, respectively. The integrability of the NLS equation was established by V. E. Zakharov and A. B. Shabat in 1972. The most powerful method for constructing spatially periodic (quasiperiodic) solutions of soliton equations is the finite-gap technique. The development of this approach was initiated by S. P. Novikov in 1974. However, as Novikov pointed out, despite the apparent simplicity of the $\Theta $-function formulas, one usually needs to make them more explicit in order to apply them. Fortunately, as noticed by P. M. Santini and the present author, in the problem of generation of rogue waves due to modulation instability, the Cauchy data at the initial time have a special form and are a small perturbation of the unstable background. In this case the spectral curves in the finite-gap approach turn out to be small perturbations of rational curves, and one can obtain very simple asymptotic formulas that surprisingly well agree with the results of numerical integration for a small number of unstable modes. The present paper provides an overview of the results obtained in this direction.

Détail

Mots-clés : rogue (anomalous) waves
Keywords: exactly solvable models, spatially periodic problem, finite-gap integration, spectral curves close to degenerate ones, asymptotic solutions in terms of elementary functions.

Bibliographie

[1] Ablowitz M.J., Hammack J., Henderson D., Schober C.M., “Long-time dynamics of the modulational instability of deep water waves”, Physica D, 152–153 (2001), 416–433 | DOI | MR | Zbl

[2] Ablowitz M.J., Herbst B.M., “On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation”, SIAM J. Appl. Math., 50:2 (1990), 339–351 | DOI | MR | Zbl

[3] Ablowitz M.J., Schober C., Herbst B.M., “Numerical chaos, roundoff errors, and homoclinic manifolds”, Phys. Rev. Lett., 71:17 (1993), 2683–2686 | DOI

[4] Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, SIAM Stud. Appl. Math., 4, SIAM, Philadelphia, 1981 | DOI | MR | Zbl

[5] Agafontsev D.S., Gelash A.A., Mullyadzhanov R.I., Zakharov V.E., “Bound-state soliton gas as a limit of adiabatically growing integrable turbulence”, Chaos Solitons Fractals, 166 (2023), 112951 | DOI | MR

[6] Agafontsev D.S., Zakharov V.E., “Integrable turbulence and formation of rogue waves”, Nonlinearity, 28:8 (2015), 2791–2821 | DOI | MR | Zbl

[7] Akhmediev N.N., “Déjà vu in optics”, Nature, 413 (2001), 267–268 | DOI

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

[9] N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: Exact solutions”, Sov. Phys., JETP, 62:5 (1985), 894–899

[10] N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schrödinger equation”, Theor. Math. Phys., 72:2 (1987), 809–818 | DOI | MR | Zbl

[11] N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation”, Theor. Math. Phys., 69:2 (1986), 1089–1093 | DOI | MR | Zbl

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

[13] Ayano T., Buchstaber V.M., “Relationships between hyperelliptic functions of genus 2 and elliptic functions”, SIGMA, Symmetry Integrability Geom. Methods Appl., 18 (2022), 010 | DOI | MR | Zbl

[14] M. V. Babich, A. I. Bobenko, and V. B. Matveev, “Reductions of Riemann theta-functions of genus $g$ to theta-functions of lower genus, and symmetries of algebraic curves”, Sov. Math., Dokl., 28 (1983), 304–308 | MR | Zbl

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

[16] 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 Dyn., Springer, Berlin, 1994 | Zbl

[17] Benjamin T.B., Feir J.E., “The disintegration of wave trains on deep water. Part I: Theory”, J. Fluid Mech., 27:3 (1967), 417–430 | DOI | Zbl

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

[19] Bertola M., El G.A., Tovbis A., “Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation”, Proc. R. Soc. London A: Math. Phys. Eng. Sci., 472:2194 (2016), 20160340 | DOI | MR | Zbl

[20] Bertola M., Tovbis A., “Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquée solution to Painlevé I”, Commun. Pure Appl. Math., 66:5 (2013), 678–752 | DOI | MR | Zbl

[21] V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids”, JETP Lett., 3:12 (1966), 307–310

[22] Biondini G., Kireyev D., “On the periodic solutions of the Davey–Stewartson systems”, East Asian J. Appl. Math., 12 (2022), 1–21 | DOI | MR | Zbl

[23] Biondini G., Kireyev D., Maruno K., “Soliton resonance and web structure in the Davey–Stewartson system”, J. Phys. A: Math. Theor., 55:30 (2022), 305701 | DOI | MR | Zbl

[24] Biondini G., Li S., Mantzavinos D., “Oscillation structure of localized perturbations in modulationally unstable media”, Phys. Rev. E, 94:6 (2016), 060201(R) | DOI

[25] Biondini G., Luo X.-D., Oregero J., Tovbis A., “Elliptic finite-band potentials of a non-self-adjoint Dirac operator”, Adv. Math., 429 (2023), 109188 | DOI | MR | Zbl

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

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

[28] Bortolozzo U., Montina A., Arecchi F.T., Huignard J.P., Residori S., “Spatiotemporal pulses in a liquid crystal optical oscillator”, Phys. Rev. Lett., 99:2 (2007), 023901 | DOI

[29] Boutet de Monvel A., Kotlyarov V.P., Shepelsky D., “Focusing NLS equation: Long-time dynamics of step-like initial data”, Int. Math. Res. Not., 2011:7 (2011), 1613–1653 | DOI | MR | Zbl

[30] Buckingham R., Venakides S., “Long-time asymptotics of the nonlinear Schrödinger equation shock problem”, Commun. Pure Appl. Math., 60:9 (2007), 1349–1414 | DOI | MR | Zbl

[31] Calini A., Ercolani N.M., McLaughlin D.W., Schober C.M., “Mel'nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation”, Physica D, 89:3–4 (1996), 227–260 | DOI | MR | Zbl

[32] Calini A., Schober C.M., “Homoclinic chaos increases the likelihood of rogue wave formation”, Phys. Lett. A, 298:5–6 (2002), 335–349 | DOI | MR | Zbl

[33] Calini A., Schober C.M., “Dynamical criteria for rogue waves in nonlinear Schrödinger models”, Nonlinearity, 25:12 (2012), R99–R116 | DOI | MR | Zbl

[34] Calini A., Schober C.M., “Observable and reproducible rogue waves”, J. Opt., 15:10 (2013), 105201 | DOI

[35] Calini A., Schober C.M., “Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation”, Nat. Hazards Earth Syst. Sci., 14:6 (2014), 1431–1440 | DOI

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

[37] I. V. Cherednik, “Reality conditions in ‘finite-zone integration’”, Sov. Phys., Dokl., 25:6 (1980), 450–452 | MR | Zbl

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

[39] Coppini F., Grinevich P.G., Santini P.M., “Periodic rogue waves and perturbation theory”, Perturbation theory: Mathematics, methods and applications, Encycl. Complex. Syst. Sci. Ser., Springer, New York, 2022, 565–584 | DOI | MR

[40] Coppini F., Grinevich P.G., Santini P.M., “The periodic $N$ breather anomalous wave solution of the Davey–Stewartson equations; first appearance, recurrence, and blow up properties”, J. Phys. A: Math. Theor., 57:1 (2023), 015208 | DOI | MR

[41] Coppini F., Santini P.M., “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 | DOI | MR

[42] Coppini F., Santini P.M., “Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz–Ladik lattices”, J. Phys. A: Math. Theor., 57:1 (2024), 015202 | DOI | MR | Zbl

[43] Coppini F., Santini P.M., “The effect of loss/gain and Hamiltonian perturbations of the Ablowitz–Ladik lattice on the recurrence of periodic anomalous waves”, J. Phys. A: Math. Theor., 57:7 (2024), 075701 | DOI | MR | Zbl

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

[45] Deift P., Zhou X., “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation”, Ann. Math., 137:2 (1993), 295–368 | DOI | MR | Zbl

[46] Dematteis G., Grafke T., Onorato M., Vanden-Eijnden E., “Experimental evidence of hydrodynamic instantons: The universal route to rogue waves”, Phys. Rev. X, 9:4 (2019), 041057 | DOI

[47] Dubard P., Gaillard P., Klein C., Matveev V.B., “On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation”, Eur. Phys. J. Spec. Top., 185 (2010), 247–258 | DOI

[48] B. A. Dubrovin, “Inverse problem for periodic finite-zoned potentials in the theory of scattering”, Funct. Anal. Appl., 9:1 (1975), 61–62 | DOI | MR | Zbl

[49] B. A. Dubrovin, “Theta functions and non-linear equations”, Russ. Math. Surv., 36:2 (1981), 11–92 | DOI | MR | Zbl | Zbl

[50] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “The Schrödinger equation in a periodic field and Riemann surfaces”, Sov. Math., Dokl., 17 (1976), 947–951 | MR | Zbl

[51] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties”, Russ. Math. Surv., 31:1 (1976), 59–146 | DOI | MR | Zbl | Zbl

[52] Dyachenko A.I., Kachulin D.I., Zakharov V.E., “Freak-waves: Compact equation versus fully nonlinear one”, Extreme ocean waves, Springer, Cham, 2016, 23–44 | DOI

[53] Dyachenko A.I., Kachulin D.I., Zakharov V.E., “Super compact equation for water waves”, J. Fluid Mech., 828 (2017), 661–679 | DOI | MR | Zbl

[54] A. I. Dyachenko and V. E. Zakharov, “Modulation instability of Stokes wave $\to $ freak wave”, JETP Lett., 81:6 (2005), 255–259 | DOI

[55] A. I. Dyachenko and V. E. Zakharov, “On the formation of freak waves on the surface of deep water”, JETP Lett., 88:5 (2008), 307–311 | DOI

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

[57] Efimov V.B., Ganshin A.N., Kolmakov G.V., McClintock P.V.E., Mezhov-Deglin L.P., “Rogue waves in superfluid helium”, Eur. Phys. J. Spec. Top., 185 (2010), 181–193 | DOI

[58] El G.A., “Soliton gas in integrable dispersive hydrodynamics”, J. Stat. Mech. Theory Exp., 2021 (2021), 114001 | DOI | MR

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

[60] Ercolani N., Forest M.G., McLaughlin D.W., “Geometry of the modulational instability. III: Homoclinic orbits for the periodic sine-Gordon equation”, Physica D, 43:2–3 (1990), 349–384 | DOI | MR | Zbl

[61] Fay J.D., Theta functions on Riemann surfaces, Lect. Notes Math., 352, Springer, Berlin, 1973 | DOI | MR | Zbl

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

[63] Fermi E., Pasta J., Ulam S., Studies of non linear problems, Document LA-1940, Los Alamos Natl. Lab., Los Alamos, 1955; Fermi E., Note e memorie (Collected papers), Univ. Chicago Press, Chicago, 1965, 977–988 | MR

[64] Ganshin A.N., Efimov V.B., Kolmakov G.V., Mezhov-Deglin L.P., McClintock P.V.E., “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium”, Phys. Rev. Lett., 101:6 (2008), 065303 | DOI

[65] Gelash A.A., “Formation of rogue waves from a locally perturbed condensate”, Phys. Rev. E, 97:2 (2018), 022208 | DOI

[66] Gelash A., Agafontsev D., Zakharov V., et al., “Bound state soliton gas dynamics underlying the spontaneous modulational instability”, Phys. Rev. Lett., 123:23 (2019), 234102 | DOI | MR

[67] Gelash A., Xu G., Kibler B., “Management of breather interactions”, Phys. Rev. Res., 4:3 (2022), 033197 | DOI | MR

[68] Gelash A.A., Zakharov V.E., “Superregular solitonic solutions: A novel scenario for the nonlinear stage of modulation instability”, Nonlinearity, 27:4 (2014), R1–R39 | DOI | MR

[69] Gelfand I.M., Levitan B.M., “Ob opredelenii differentsialnogo uravneniya po ego spektralnoi funktsii”, Izv. AN SSSR. Ser. mat., 15:4 (1951), 309–360 | Zbl

[70] Grinevich P.G., Novikov S.P., “Topological charge of the real periodic finite-gap sine-Gordon solutions”, Commun. Pure Appl. Math., 56:7 (2003), 956–978 | DOI | MR | Zbl

[71] Grinevich P.G., Santini P.M., “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

[72] Grinevich P.G., Santini P.M., “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

[73] Grinevich P.G., Santini P.M., “Numerical instability of the Akhmediev breather and a finite-gap model of it”, Recent developments in integrable systems and related topics of mathematical physics (Kezenoi-Am, Russia, 2016), Springer Proc. Math. Stat., 273, ed. by V.M. Buchstaber et al., Springer, Cham, 2018, 3–23 | DOI | Zbl

[74] P. G. Grinevich and P. M. Santini, “Phase resonances of the NLS rogue wave recurrence in the quasisymmetric case”, Theor. Math. Phys., 196:3 (2018), 1294–1306 | DOI | DOI | MR | Zbl

[75] 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”, Russ. Math. Surv., 74:2 (2019), 211–263 | DOI | DOI | MR | Zbl

[76] Grinevich P.G., Santini P.M., “The linear and nonlinear instability of the Akhmediev breather”, Nonlinearity, 34:12 (2021), 8331–8358 | DOI | MR | Zbl

[77] P. G. Grinevich and P. M. Santini, “The finite-gap method and the periodic Cauchy problem for $(2+1)$-dimensional anomalous waves for the focusing Davey–Stewartson 2 equation”, Russ. Math. Surv., 77:6 (2022), 1029–1059 | DOI | DOI | MR | Zbl

[78] Grinevich P.G., Schmidt M.U., “Period preserving nonisospectral flows and the moduli space of periodic solutions of soliton equations”, Physica D, 87:1–4 (1995), 73–98 | DOI | MR | Zbl

[79] Hammani K., Wetzel B., Kibler B., et al., “Spectral dynamics of modulation instability described using Akhmediev breather theory”, Opt. Lett., 36:11 (2011), 2140–2142 | DOI

[80] Haver S., Freak wave event at Draupner jacket January 1, 1995, Tech. Rep. PTT-KU-MA, Statoil, Stavanger, 2003 https://home.itp.ac.ru/~alexd/HSE/Haver2004.pdf

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

[82] Herbst B.M., Ablowitz M.J., “Numerically induced chaos in the nonlinear Schrödinger equation”, Phys. Rev. Lett., 62:18 (1989), 2065–2068 | DOI | MR

[83] Hirota R., “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27:18 (1971), 1192–1994 ; Khirota R., “Pryamye metody v teorii solitonov”, Solitony, Pod. red. R. Bullafa, F. Kodri, Mir, M., 1983, 175–192 | DOI

[84] R. Hirota, “Direct methods in soliton theory”, Solitons, Top. Curr. Phys., 17, ed. by R. K. Bullough and P. J. Caudrey, Springer, Berlin, 1980, 157–176 | DOI | MR

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

[86] Its A.R., Kotlyarov V.P., “Yavnye formuly dlya reshenii nelineinogo uravneniya Shredingera”, DAN USSR. Ser. A, 1976, no. 11, 965–968 | Zbl

[87] A. R. Its and V. B. Matveev, “Hill's operator with finitely many gaps”, Funct. Anal. Appl., 9:1 (1975), 65–66 | DOI | MR | Zbl

[88] A. R. Its, A. V. Rybin, and M. A. Sall', “Exact integration of nonlinear Schrödinger equation”, Theor. Math. Phys., 74:1 (1988), 20–32 | DOI | MR | Zbl

[89] Kachulin D., Dyachenko A., Zakharov V., “Soliton turbulence in approximate and exact models for deep water waves”, Fluids, 5:2 (2020), 67 | DOI

[90] Kawata T., Inoue H., “Inverse scattering method for the nonlinear evolution equations under nonvanishing conditions”, J. Phys. Soc. Japan, 44:5 (1978), 1722–1729 | DOI | MR | Zbl

[91] Kedziora D.J., Ankiewicz A., Akhmediev N., “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits”, Phys. Rev. E, 85:6 (2012), 066601 | DOI

[92] Kharif C., Pelinovsky E., Slunyaev A., Rogue waves in the ocean, Springer, Berlin, 2009 | DOI | MR | Zbl

[93] Kibler B., Fatome J., Finot C., et al., “The Peregrine soliton in nonlinear fibre optics”, Nature Phys., 6:10 (2010), 790–795 | DOI

[94] Kibler B., Fatome J., Finot C., et al., “Observation of Kuznetsov–Ma soliton dynamics in optical fibre”, Sci. Rep., 2 (2012), 463 | DOI

[95] Kimmoun O., Hsu H.C., Branger H., et al., “Modulation instability and phase-shifted Fermi–Pasta–Ulam recurrence”, Sci. Rep., 6 (2016), 28516 | DOI

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

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

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

[99] Konopelchenko B.G., “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

[100] I. M. Krichever, “Potentials with zero coefficient of reflection on a background of finite-zone potentials”, Funct. Anal. Appl., 9:2 (1975), 161–163 | DOI | MR | Zbl

[101] I. M. Kričever, “Algebraic-geometric construction of the Zaharov–Šabat equations and their periodic solutions”, Sov. Math., Dokl., 17 (1976), 394–397 | MR | Zbl

[102] I. M. Krichever, “Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model”, Funct. Anal. Appl., 20:3 (1986), 203–214 | DOI | MR | Zbl

[103] I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications”, Russ. Math. Surv., 44:2 (1989), 145–225 | DOI | MR | Zbl

[104] Krichever I.M., “Perturbation theory in periodic problems for two-dimensional integrable systems”, Sov. Sci. Rev. Sect. C, 9:2 (1992), 1–103 | MR | Zbl

[105] E. A. Kuznetsov, “Solitons in a parametrically unstable plasma”, Sov. Phys., Dokl., 22 (1977), 507–508

[106] E. A. Kuznetsov, “Fermi–Pasta–Ulam recurrence and modulation instability”, JETP Lett., 105:2 (2017), 125–129 | DOI | DOI

[107] Lake B.M., Yuen H.C., Rungaldier H., Ferguson W.E., “Nonlinear deep-water waves: Theory and experiment. Part 2: Evolution of a continuous wave train”, J. Fluid Mech., 83:1 (1977), 49–74 | DOI | MR

[108] Lax P.D., “Periodic solutions of the KdV equation”, Commun. Pure Appl. Math., 28 (1975), 141–188 | DOI | MR | Zbl

[109] Li Y., McLaughlin D.W., “Homoclinic orbits and chaos in discretized perturbed NLS systems. Part I: Homoclinic orbits”, J. Nonlinear Sci., 7:3 (1997), 211–269 | DOI | MR | Zbl

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

[111] Liu C., van der Wel R.E.C., Rotenberg N., et al., “Triggering extreme events at the nanoscale in photonic seas”, Nature Phys., 11:4 (2015), 358–363 | DOI

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

[113] Ma Y.-C., “The perturbed plane-wave solutions of the cubic Schrödinger equation”, Stud. Appl. Math., 60:1 (1979), 43–58 | DOI | MR | Zbl

[114] S. V. Manakov, “Nonlinear Fraunhofer diffraction”, Sov. Phys., JETP, 38:4 (1974), 693–696

[115] Manakov S.V., “Metod obratnoi zadachi rasseyaniya i dvumernye evolyutsionnye uravneniya”, UMN, 31:5 (1976), 245–246 | MR | Zbl

[116] Marchenko V.A., “Nekotorye voprosy teorii odnomernykh lineinykh differentsialnykh operatorov vtorogo poryadka. I”, Tr. Mosk. mat. o-va, 1 (1952), 327–420

[117] V. A. Marčenko and I. V. Ostrovskiĭ, “A characterization of the spectrum of Hill's operator”, Math. USSR, Sb., 26:4 (1975), 493–554 | DOI | MR | MR | Zbl

[118] 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”, Math. Notes, 100:5–6 (2016), 835–846 | DOI | DOI | MR | Zbl

[119] Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Ser. Nonlinear Dyn., Springer, Berlin, 1991 | DOI | MR | Zbl

[120] V. B. Matveev and A. O. Smirnov, “Elliptic solitons and ‘freak waves’”, St. Petersburg Math. J., 33:3 (2022), 523–551 | DOI | MR | Zbl

[121] McKean H.P., van Moerbeke P., “The spectrum of Hill's equation”, Invent. math., 30:3 (1975), 217–274 | DOI | MR | Zbl

[122] McLaughlin D.W., “Whiskered tori for NLS equations”, Important developments in soliton theory, Springer Ser. Nonlinear Dyn., ed. by A.S. Fokas, V.E. Zakharov, Springer, Berlin, 1993, 537–558 | DOI | MR | Zbl

[123] McLaughlin D.W., Overman E.A., \textup {II.}, “Whiskered tori for integrable PDE's: Chaotic behavior in near integrable PDE's”, Surveys in applied mathematics, v. 1, ed. by J.B. Keller et al., Plenum Press, New York, 1995, 83–203 | DOI | MR | Zbl

[124] McLaughlin D.W., Schober C.M., “Chaotic and homoclinic behavior for numerical discretizations of the nonlinear Schrödinger equation”, Physica D, 57:3–4 (1992), 447–465 | DOI | MR | Zbl

[125] Moslem W.M., Sabry R., El-Labany S.K., Shukla P.K., “Dust-acoustic rogue waves in a nonextensive plasma”, Phys. Rev. E, 84:6 (2011), 066402 ; Mamford D., Lektsii o teta-funktsiyakh, Mir, M., 1988 | DOI | MR

[126] D. Mumford, Tata Lectures on Theta. I, Prog. Math., 28, Birkhäuser, Boston, 1983 | DOI | MR | MR | Zbl

[127] Naveau C., Szriftgiser P., Kudlinski A., et al., “Experimental characterization of recurrences and separatrix crossing in modulational instability”, Opt. Lett., 44:22 (2019), 5426–5429 | DOI

[128] Nguyen J., Dyke P., Luo D., Malomed B.A., Hullet R., “Collisions of matter-wave solitons”, Nature Phys., 10:12 (2014), 918–922 | DOI

[129] K. Nishinari, K. Abe, and J. Satsuma, “A new type of soliton behavior of the Davey–Stewartson equations in a plasma system”, Theor. Math. Phys., 99:3 (1994), 745–753 | DOI | MR | Zbl

[130] S. P. Novikov, “The periodic problem for the Korteweg–de Vries equation”, Funct. Anal. Appl., 8:3 (1974), 236–246 | DOI | MR | Zbl

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

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

[133] Onorato M., Residori S., Bortolozzo U., Montina A., Arecchi F.T., “Rogue waves and their generating mechanisms in different physical contexts”, Phys. Rep., 528:2 (2013), 47–89 | DOI | MR

[134] Onorato M., Waseda T., Toffoli A., et al., “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

[135] Osborne A.R., Onorato M., Serio M., “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

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

[137] Pedit F., Pinkall U., “Quaternionic analysis on Riemann surfaces and differential geometry”, Doc. math. Extra Vol. ICM Berlin, II (1998), 389–400 | MR

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

[139] Pierangeli D., Di Mei F., Conti C., Agranat A.J., DelRe E., “Spatial rogue waves in photorefractive ferroelectrics”, Phys. Rev. Lett., 115:9 (2015), 093901 | DOI | MR

[140] Pierangeli D., Flammini M., Zhang L., et al., “Observation of Fermi–Pasta–Ulam–Tsingou recurrence and its exact dynamics”, Phys. Rev. X, 8:4 (2018), 041017 | DOI

[141] Segur H., Henderson D., Carter J., et al., “Stabilizing the Benjamin–Feir instability”, J. Fluid Mech., 539 (2005), 229–271 | DOI | MR | Zbl

[142] A. V. Slunyaev, D. E. Pelinovsky, and E. N. Pelinovsky, “Rogue waves in the sea: Observations, physics, and mathematics”, Phys. Usp., 66:2 (2023), 148–172 | DOI | DOI | MR

[143] A. O. Smirnov, “Periodic two-phase ‘rogue waves’”, Math. Notes, 94:5–6 (2013), 897–907 | DOI | DOI | MR | Zbl

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

[145] Soto-Crespo J.M., Devine N., Akhmediev N., “Adiabatic transformation of continuous waves into trains of pulses”, Phys. Rev. A, 96:2 (2017), 023825 | DOI | MR

[146] Romero-Ros A., Katsimiga G.C., Mistakidis S.I., et al., “Experimental realization of the Peregrine soliton in repulsive two-component Bose–Einstein condensates”, Phys. Rev. Lett., 132:3 (2024), 033402 | DOI

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

[148] I. A. Taimanov, “On two-dimensional finite-gap potential Schrödinger and Dirac operators with singular spectral curves”, Sib. Math. J., 44:4 (2003), 686–694 | DOI | MR | Zbl

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

[150] I. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces”, Russ. Math. Surv., 61:1 (2006), 79–159 | DOI | DOI | MR | Zbl

[151] I. A. Taimanov, “Singular spectral curves in finite-gap integration”, Russ. Math. Surv., 66:1 (2011), 107–144 | DOI | DOI | MR | Zbl

[152] I. A. Taimanov, “The Moutard transformation of two-dimensional Dirac operators and Möbius geometry”, Math. Notes, 97:1–2 (2015), 124–135 | DOI | DOI | MR | Zbl

[153] I. A. Taimanov, “Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces”, Theor. Math. Phys., 182:2 (2015), 173–181 | DOI | DOI | MR | Zbl

[154] I. A. Taimanov, “The Moutard transformation for the Davey–Stewartson II equation and its geometrical meaning”, Math. Notes, 110:5–6 (2021), 754–766 | DOI | DOI | MR | Zbl

[155] I. A. Taimanov and S. P. Tsarev, “Two-dimensional rational solitons and their blowup via the Moutard transformation”, Theor. Math. Phys., 157:2 (2008), 1525–1541 | DOI | DOI | MR | Zbl

[156] Tajiri M., Watanabe Y., “Breather solutions to the focusing nonlinear Schrödinger equation”, Phys. Rev. E, 57:3 (1998), 3510–3519 | DOI | MR

[157] The Fermi–Pasta–Ulam problem: A status report, Lect. Notes Phys., 728, ed. by G. Gallavotti, Springer, Berlin, 2008 | DOI | MR | Zbl

[158] Van Simaeys G., Emplit Ph., Haelterman M., “Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave”, Phys. Rev. Lett., 87:3 (2001), 033902 | DOI

[159] Wen L., Li L., Li Z.D., et al., “Matter rogue wave in Bose–Einstein condensates with attractive atomic interaction”, Eur. Phys. J. D, 64:2–3 (2011), 473–478 | DOI

[160] Yuen H.C., Ferguson W.E., Jr., “Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation”, Phys. Fluids, 21:8 (1978), 1275–1278 | DOI

[161] Yuen H.C., Lake B.M., “Nonlinear dynamics of deep-water gravity waves”, Adv. Appl. Mech., 22 (1982), 67–229 | DOI | MR | Zbl

[162] Zabusky N.J., Kruskal M.D., “Interaction of “solitons” in a collisionless plasma and the recurrence of initial states”, Phys. Rev. Lett., 15:6 (1965), 240–243 | DOI | Zbl

[163] V. E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, J. Appl. Mech. Tech. Phys., 9:2 (1968), 190–194 | DOI

[164] V. E. Zakharov, “Collapse of Langmuir waves”, Sov. Phys., JETP, 35:5 (1972), 908–914

[165] Zakharov V.E., Dyachenko A.I., “About shape of giant breather”, Eur. J. Mech. B/Fluids, 29:2 (2010), 127–131 | DOI | MR | Zbl

[166] Zakharov V.E., Dyachenko A.I., Prokofiev A.O., “Freak waves as nonlinear stage of Stokes wave modulation instability”, Eur. J. Mech. B/Fluids, 25:5 (2006), 677–692 | DOI | MR | Zbl

[167] Zakharov V.E., Gelash A.A., Soliton on unstable condensate, E-print, 2011, arXiv: 1109.0620v2 [nlin.SI] | DOI

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

[169] V. E. Zakharov and S. V. Manakov, “Construction of higher-dimensional nonlinear integrable systems and of their solutions”, Funct. Anal. Appl., 19:2 (1985), 89–101 | DOI | MR | Zbl

[170] V. E. Zakharov and A. V. Mikhaĭlov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method”, Sov. Phys., JETP, 47:6 (1987), 1017–1027 | MR

[171] Zakharov V.E., Ostrovsky L.A., “Modulation instability: The beginning”, Physica D, 238:5 (2009), 540–548 | DOI | MR | Zbl

[172] V. E. Zakharov and A. M. Rubenchik, “Nonlinear interaction of high-frequency and low-frequency waves”, J. Appl. Mech. Tech. Phys., 13:5 (1972), 669–681 | DOI

[173] V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Sov. Phys., JETP, 34:1 (1972), 62–69 | MR | MR

[174] 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”, Funct. Anal. Appl., 8:3 (1974), 226–235 | DOI | MR | Zbl

[175] V. E. Zakharov and R. V. Shamin, “Probability of the occurrence of freak waves”, JETP Lett., 91:2 (2010), 62–65 | DOI