Finite-gap solutions of nonlocal equations in Ablowitz-Kaup-Newell-Segur hierarchy

A. O. Smirnov   ; V. B. Matveev

Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 81-98

Résumé

Nonlinear nonlocal models exist in many fields of physics. The most known of them are models possessing $\mathcal{PT}$-symmetries. Apart of $\mathcal{PT}$-symmetric models, nonlocal models with inverse time and/or coordinates are actively studied. Other types of nonlocalities arise much rare. As a rule, in works devoted to nonlinear nonlocal equations, soliton or quasi-rational solutions to such equations are studied. In the present work we consider nonlocal symmetries, to which all equations in the Ablowitz-Kaup-Newell-Segur hierarchy. On the base of the properties of solutions satisfying nonlocal reductions of the equations in the Ablowitz-Kaup-Newell-Segur hierarchy, we propose a modification of theta-functional formula for Baker-Akhiezer functions. We find the conditions for the parameters of spectral curves associated with multi-phase solutions possessing no exponential growth at infinity. We show that under these conditions, the variables separate. The most part of statement of our work remain true for soliton and quasi-rational solutions since they are limiting cases for the multi-phase solutions.

Détail

Keywords: nonlinear Schrödinger equation, Ablowitz-Kaup-Newell-Segur hierarchy, PT-symmetry, finite-gap solution, spectral curve, theta function.
Mots-clés : nonlocal equation

Bibliographie

[1] V. Konotop, J. Yang, D. Zezulin, “Nonlinear waves in PT-symmetric systems”, Rev. Modern Phys., 88 (2016), 035002 | DOI

[2] D. Christodoulides, J. Yang (eds.), Parity-time symmetry and its applications, Springer Tracts in Modern Physics, 280, Springer, 2018 | DOI

[3] M. J. Ablowitz, Z. H. Musslimani, “Integrable nonlocal nonlinear Schrödinger equation”, Phys. Rev. Let., 110 (2013), 064105 | DOI

[4] M. J. Ablowitz, Z. H. Musslimani, “Integrable discrete PT symmetric model”, Phys. Rev. E, 90:3 (2014), 032912 | DOI | MR

[5] M. J. Ablowitz, Z. H. Musslimani, “Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation”, Nonlinearity, 29:3 (2016), 915–946 | DOI | MR | Zbl

[6] M. J. Ablowitz, Z. H. Musslimani, “Integrable nonlocal nonlinear equations”, Stud. Appl. Math., 139:1 (2017), 7–59 | DOI | MR | Zbl

[7] T. P. Horikis, M. J. Ablowitz, “Rogue waves in nonlocal media”, Phys. Rev. E, 95:4 (2017), 042211 | DOI | MR

[8] M. J. Ablowitz, B. F. Feng, X. D. Luo, Z. H. Musslimani, “Reverse space-time nonlocal Sine-Gordon/Sinh-Gordon equations with nonzero boundary conditions”, Stud. Appl. Math., 2018 | MR

[9] V. S. Gerdjikov, G. G. Grahovski, R. I. Ivanov, “The $N$-wave equations with PT symmetry”, Theor. Math. Phys., 188:3 (2016), 1305–1321 | DOI | MR | Zbl

[10] D. Y. Liu, W. R. Sun, “Rational solutions for the nonlocal sixth-order nonlinear Schrödinger equation”, Appl. Math. Lett., 84 (2018), 63–69 | DOI | MR | Zbl

[11] H. Q. Zhang, M. Gao, “Rational soliton solutions in the parity-time-symmetric nonlocal coupled nonlinear Schrödinger equations”, Comm. Nonlin. Sci. and Num. Sim., 63 (2018), 253–260 | DOI | MR | Zbl

[12] Z. X. Zhou, “Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrodinger equation”, Commun Nonlinear Sci. Numer. Simulat., 62 (2018), 480–488 | DOI | MR | Zbl

[13] Y. Cao, B. Malomed, J. He, “Two (2+1)-dimensional integrable nonlocal nonlinear Schrodinger equations: Breather, rational and semi-rational solutions”, Chaos, Solitons and Fractals, 114 (2018), 99–107 | DOI | MR | Zbl

[14] B. Yang, Y. Chen, “Reductions of Darboux transformations for the PT-symmetric nonlocal Davey-Stewartson equations”, Appl. Math. Lett., 82 (2018), 43–49 | DOI | MR | Zbl

[15] Z. J. Yang, S. M. Zhang, X. L. Li, Z. G. Pang, “Variable sinh-Gaussian solitons in nonlocal nonlinear Schrodinger equation”, Appl. Math. Lett., 82 (2018), 64–70 | DOI | MR | Zbl

[16] K. Manikandan, Priya N. Vishnu, M. Senthilvelan, R. Sankaranarayanan, “Deformation of dark solitons in a PT-invariant variable coefficients nonlocal nonlinear Schrodinger equation”, Chaos, 28:8 (2018), 083103 | DOI | MR | Zbl

[17] J. Rao, Y. Zhang, A. Fokas, J. He, “Rogue waves of the nonlocal Davey-Stewartson I equation”, Nonlinearity, 31:9 (2018), 4090–4107 | DOI | MR | Zbl

[18] X. Y. Tang, Z. F. Liang, X. Z. Hao, “Nonlinear waves of a nonlocal modified KdV equation in the atmospheric and oceanic dynamical system”, Commun Nonlinear Sci. Numer. Simulat., 60 (2018), 62–71 | DOI | MR | Zbl

[19] K. Chen, X. Deng, S. Lou, D. J. Zhang, “Solutions of nonlocal equations reduced from the AKNS hierarchy”, Stud. Appl. Math., 141:1 (2018), 113–141 | DOI | MR | Zbl

[20] W. Liu, X. Li, “General soliton solutions to a (2+1)-dimensional nonlocal nonlinear Schrodinger equation with zero and nonzero boundary conditions”, Nonlinear Dynamics, 93:2 (2018), 721–731 | DOI | MR | Zbl

[21] P. Vinayagam, R. Radha, U. Al Khawaja, L. Ling, “New classes of solutions in the coupled PT symmetric nonlocal nonlinear Schrodinger equations with four wave mixing”, Commun Nonlinear Sci. Numer. Simulat., 59 (2018), 387–395 | DOI | MR | Zbl

[22] Y. Cao, J. Rao, D. Mihalache, J. He, “Semi-rational solutions for the (2+1)-dimensional nonlocal Fokas system”, Appl. Math. Lett., 80 (2018), 27–34 | DOI | MR | Zbl

[23] C. Qian, J. Rao, D. Mihalache, J. He, “Rational and semi-rational solutions of the $y$-nonlocal Davey-Stewartson I equation”, Computers and Mathematics with Applications, 75:9 (2018), 3317–3330 | DOI | MR | Zbl

[24] M. Gurses, A. Pekcan, “Nonlocal modified KdV equations and their soliton solutions by Hirota method”, Commun Nonlinear Sci. Numer. Simulat., 67 (2019), 427–448 | DOI | MR | Zbl

[25] Q. Zhang, Y. Zhang, R. Ye, “Exact solutions of nonlocal Fokas-Lenells equation”, Appl. Math. Lett., 98 (2019), 336–343 | DOI | MR | Zbl

[26] V. S. Gerdjikov, “On the integrability of Ablowitz-Ladik models with local and nonlocal reductions”, Journal of Physics: Conference Series, 1205:1 (2019), 012015 | DOI

[27] B. Yang, J. Yang, “Rogue waves in the nonlocal PT-symmetric nonlinear Schrödinger equation”, Lett. Math. Phys., 109 (2019), 945–973 | DOI | MR | Zbl

[28] J. Yang, “General $n$-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations”, Phys. Lett. A, 383 (2019), 328–337 | DOI | MR

[29] B. Yang, J. Yang, On general rogue waves in the parity-time-symmetric nonlinear Schrodinger equation, 2019, 19 pp., arXiv: 1903.06203 | MR

[30] Y. Yang, T. Suzuki, X. Cheng, “Darboux transformations and exact solutions for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation”, Appl. Math. Lett., 99 (2020), 105998 | DOI | MR | Zbl

[31] A. O. Smirnov, E. E. Aman, “The simplest oscillating solutions of nonlocal nonlinear models”, Journal of Physics: Conference Series, 1399:2 (2019), 022020 | DOI

[32] A. O. Smirnov, E. E. Aman, “One-phase elliptic solutions of the nonlocal nonlinear equations from AKNS hierarchy and their spectral curves”, Journal of Physics: Conference Series, 1515:3 (2020), 032080 | DOI

[33] V. B. Matveev, A. O. Smirnov, “Multiphase solutions of nonlocal symmetric reductions of equations of the AKNS hierarchy: general analysis and simplest examples”, Theor. Math. Phys., 204:3 (2020), 1154–1165 | DOI | MR | Zbl

[34] V. B. Matveev, A. O. Smirnov, “AKNS hierarchy, MRW solutions, $P_n$ breathers, beyond”, J. Math. Phys., 59:9 (2018), 091419 | DOI | MR | Zbl

[35] V. B. Matveev, A. O. Smirnov, “Two-phase periodic solutions to the AKNS hierarchy equations”, J. Math. Sci., 242:5 (2019), 722–741 | DOI | MR | Zbl

[36] A. R. Its, 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) | MR

[37] V. P. Kotlyarov, Periodic problem for the nonlinear Schrödinger equation, 2014, 14 pp., arXiv: 1401.4445 | Zbl

[38] 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

[39] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems”, Studies in Appl. Math., 53:4 (1974), 249–315 | DOI | MR | Zbl

[40] 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

[41] 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

[42] 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

[43] A.O Smirnov, “Solution of a nonlinear Schrödinger equation in the form of two-phase freak waves”, Theor. Math. Phys., 173:1 (2012), 1403–1416 | DOI | MR | Zbl

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

[45] A.O Smirnov, S. G. Matveenko, S. K. Semenov, E. G. Semenova, “Three-phase freak waves”, SIGMA, 11 (2015), 032 | MR | Zbl

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

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

[48] A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903 | MR | Zbl

[49] H. F. Baker, Abel's theorem and the allied theory including the theory of theta functions, Cambridge, 1897 | MR

[50] D. Mumford, Tata lectures on theta, v. I, Progress in Math., 28, Birkhäuser Boston Inc., Boston, MA, 1983 | DOI | MR | Zbl

[51] D. Mumford, Tata lectures on theta, v. II, Progress in Math., 43, Birkhäuser Boston Inc., Boston, MA, 1984 | MR | Zbl

[52] A. O. Smirnov, “A matrix analogue of the Appell theorem and reduction of multidimensional Riemann theta-functions”, Math. USSR Sb., 61:2 (1988), 379–388 | DOI | MR | Zbl