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.

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