Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

Abstract

We investigate the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with a particular class of nonvanishing boundary conditions (NVBCs), especially in the case of reflectionless potentials that give rise to a transmission coefficient with an arbitrary finite number N pairs of higher-order poles. The inverse problem is characterized in terms of a

2\times 2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2\times 2$$\end{document}

matrix Riemann-Hilbert (RH) problem equipped with several residue conditions at N pairs of higher-order poles. In the reflectionless case, we point out that the N-multipole soliton solutions including higher-order Kuznetsov-Ma breathers and Akhmediev breathers can be reconstructed by a linear algebraic system. Furthermore, we verify these special solutions by numerical simulations and display their density structures, also derive some Peregrine solitons by choosing appropriate parameters and taking limits.

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Keywords

Nonlinear Schrödinger (NLS) equation / Nonvanishing boundary conditions (NVBCs) / Riemann-Hilbert (RH) problem / N-multipole soliton solutions / 37K10

References

[1] Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. In: London Mathematical Society Lecture Note Series, vol. 149. Cambridge University Press, Cambridge (1991)

[2] Ablowitz, M.J., Fokas, A.S.: Complex variables: introduction and applications. In: Cambridge Texts in Applied Mathematics, 2nd edn. Cambridge University Press, Cambridge (2003)

[3] Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. In: SIAM Studies in Applied Mathematics, vol. 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1981)

[4] Aktosun T, Demontis F, van der Mee C. Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Prob., 2007, 23: 2171-2195

[5] Beals R, Coifman RR. Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math., 1984, 37: 39-90

[6] Beals, R., Deift, P., Tomei, C.: Direct and inverse scattering on the line. In: Mathematical Surveys and Monographs, vol. 28. American Mathematical Society, Providence (1988)

[7] Bilman D, Miller PD. A robust inverse scattering transform for the focusing nonlinear Schrödinger equation. Commun. Pure Appl. Math., 2019, 72: 1722-1805

[8] Biondini G, Fagerstrom E. The integrable nature of modulational instability. SIAM J. Appl. Math., 2015, 75: 136-163

[9] Biondini G, Kovačič G. Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys., 2014, 55 031506

[10] Biondini G, Li S, Mantzavinos D. Soliton trapping, transmission and wake in modulationally unstable media. Phys. Rev. E, 2018, 98 042211

[11] Biondini G, Li S, Mantzavinos D. Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions in the presence of a discrete spectrum. Commun. Math. Phys., 2021, 382: 1495-1577

[12] Biondini G, Li S, Mantzavinos D, Trillo S. Universal behavior of modulationally unstable media. SIAM Rev., 2018, 60: 888-908

[13] Biondini G, Lottes J, Mantzavinos D. Inverse scattering transform for the focusing nonlinear Schrödinger equation with counterpropagating flows. Stud. Appl. Math., 2021, 146: 371-439

[14] Chen MS, Fan EG. Riemann-Hilbert approach for discrete sine-Gordon equation with simple and double poles. Stud. Appl. Math., 2022, 148: 1180-1207

[15] Chen MS, Fan EG, He JS. Riemann-Hilbert approach and the soliton solutions of the discrete mKdV equations. Chaos Solitons Fractals, 2023, 168 113209

[16] Demontis F, Prinari B, van der Mee C, Vitale F. The inverse scattering transform for the defocusing nonlinear Schrödinger equations with nonzero boundary conditions. Stud. Appl. Math., 2013, 131: 1-40

[17] Fokas, A.S.: A unified approach to boundary value problems. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)

[18] Fokas AS, Its AR. The linearization of the initial-boundary value problem of the nonlinear Schröinger equation. SIAM J. Math. Anal., 1996, 27: 738-764

[19] Kawata T, Inoue H. Inverse scattering method for the nonlinear evolution equations under nonvanishing conditions. J. Phys. Soc. Jpn., 1978, 44: 1722-1729

[20] Kibler B, Fatome J, Finot C, Millot G, Dias F, Genty G, Akhmediev N, Dudley JM. The Peregrine soliton in nonlinear fibre optics. Nat. Phys., 2010, 6: 790-795

[21] Liu H, Shen J, Geng XG. Inverse scattering transformation for the N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N$$\end{document}-component focusing nonlinear Schrödinger equation with nonzero boundary conditions. Lett. Math. Phys., 2023, 113: 23

[22] Lyu C, Liu QP. Multiple higher-order pole solutions of modified complex short pulse equation. Appl. Math. Lett., 2023, 141 108518

[23] Lyu, C., Qiu, D.Q., Liu, Q.P.: Riemann-Hilbert approach to two-component modified short-pulse system and its nonlocal reductions. Chaos 32, 093120 (2022)

[24] Ma YC. The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math., 1979, 60: 43-58

[25] Olmedilla E. Multiple pole solutions of the nonlinear Schrödinger equation. Physica D, 1987, 25: 330-346

[26] Pichler M, Biondini G. On the focusing non-linear Schrödinger equation with non-zero boundary conditions and double poles. IMA J. Appl. Math., 2017, 82: 131-151

[27] Prinari B, Ablowitz MJ, Biondini G. Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions. J. Math. Phys., 2006, 47 063508

[28] Schiebold C. Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation. Nonlinearity, 2017, 30: 2930-2981

[29] Shchesnovich VS, Yang JK. General soliton matrices in the Riemann-Hilbert problem for integrable nonlinear equations. J. Math. Phys., 2003, 44: 4604-4639

[30] Shchesnovich VS, Yang JK. Higher-order solitons in the N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N$$\end{document}-wave system. Stud. Appl. Math., 2003, 110: 297-332

[31] Tsuru H, Wadati M. The multiple pole solutions of the sine-Gordon equation. J. Phys. Soc. Jpn., 1984, 53: 2908-2921

[32] Wadati M, Ohkuma K. Multiple-pole solutions of the modified Korteweg-de Vries equation. J. Phys. Soc. Jpn., 1982, 51: 2029-2035

[33] Zakharov VE, Ostrovsky LA. Modulation instability: the beginning. Physica D, 2009, 238: 540-548

[34] Zakharov VE, Shabat AB. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 1972, 34: 62-69

[35] Zhang GQ, Yan ZY. The derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions: inverse scattering transforms and n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n$$\end{document}-double-pole solutions. J. Nonlinear Sci., 2020, 30: 3089-3127

[36] Zhang YS, Tao XX, Xu SW. The bound-state soliton solutions of the complex modified KdV equation. Inverse Prob., 2020, 36 065003

[37] Zhang YS, Tao XX, Yao TT, He JS. The regularity of the multiple higher-order poles solitons of the NLS equation. Stud. Appl. Math., 2020, 145: 812-827

[38] Zhou X. The Riemann-Hilbert problem and inverse scattering. SIAM J. Math. Anal., 1989, 20: 966-986

Funding

National Natural Science Foundation of China(12171439)

RIGHTS & PERMISSIONS

Shanghai University