Geodesic
Riemann–Hilbert problem for the Kundu-type nonlinear Schrödinger equation with $N$ distinct arbitrary-order poles
Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 1, pp. 23-43 Cet article a éte moissonné depuis la source Math-Net.Ru

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We use the Riemann–Hilbert (RH) method to study the Kundu-type nonlinear Schrödinger (Kundu–NLS) equation with a zero boundary condition in the case where the scattering coefficient has $N$ distinct arbitrary-order poles. We perform a spectral analysis of the Lax pair and consider the asymptotic property, symmetry, and analyticity of the Jost solution. Based on these results, we formulate the RH problem whose solution allows solving the considered Kundu–NLS equation. In addition, using graphic analysis, we study the characteristics of soliton solutions of some particular cases of the problem with $N$ distinct arbitrary-order poles.
Keywords: Kundu–nonlinear Schrödinger equation, zero boundary condition, Riemann–Hilbert problem, arbitrary-order pole, scattering coefficient
Mots-clés : soliton solution. 
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     author = {Zi-Yi Wang and Shou-Fu Tian and Xiao-Fan Zhang},
     title = {Riemann{\textendash}Hilbert problem for {the~Kundu-type} nonlinear {Schr\"odinger} equation with $N$ distinct arbitrary-order poles},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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Zi-Yi Wang; Shou-Fu Tian; Xiao-Fan Zhang. Riemann–Hilbert problem for the Kundu-type nonlinear Schrödinger equation with $N$ distinct arbitrary-order poles. Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 1, pp. 23-43. http://geodesic.mathdoc.fr/item/TMF_2021_207_1_a1/

[1] G. P. Agraval, Nelineinaya volokonnaya optika, Mir, M., 1996

[2] D. J. Benney, “A general theory for interactions between short and long waves”, Stud. Appl. Math., 56:1 (1976), 81–94 | DOI | MR

[3] T. Kakutani, K. Michihiro, “Marginal state of modulational instability – note of Benjamin–Feir instability”, J. Phys. Soc. Japan., 52:12 (1983), 4129–4137 | DOI

[4] H. Bailung, Y. Nakamura, “Observation of modulational instability in a multi-component plasma with negative ions”, J. Plasma Phys., 50:2 (1993), 231–242 | DOI

[5] Yu. S. Kivshar, G. P. Agraval, Opticheskie solitony. Ot volokonnykh svetovodov do fotonnykh kristallov, Fizmatlit, M., 2005

[6] B. A. Malomed, D. Mihalache, F. Wise, L. Torner, “Spatiotemporal optical solitons”, J. Opt. B, 7:5 (2005), R53–R72 | DOI

[7] L. Pitaevskii, S. Stringari, Bose–Einstein Condensation and Superfluidity, International Series of Monographs on Physics, 164, Oxford Univ. Press, Oxford, 2016 | Zbl

[8] G. Fanjoux, J. Michaud, H. Maillotte, T. Sylvestre, “Cascaded Raman slow light and optical spatial solitons in Kerr media”, Phys. Rev. A., 87:3 (2013), 033838, 6 pp. | DOI

[9] M. Li, J.-H. Xiao, W.-J. Liu, P. Wang, B. Qin, B. Tian, “Mixed-type vector solitons of the $N$-coupled mixed derivative nonlinear Schrödinger equations form optical fibers”, Phys. Rev. E, 87:3 (2013), 032914, 11 pp. | DOI

[10] F. G. Mertens, N. R. Quintero, A. R. Bishop, “Nonlinear Schrödinger solitons oscillate under a constant external force”, Phys. Rev. E, 87:3 (2013), 032917, 8 pp. | DOI

[11] J. T. Cole, Z. H. Musslimani, “Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential”, Phys. Rev. A, 90:1 (2014), 013815, 14 pp. | DOI

[12] A. Kundu, “Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations”, J. Math. Phys., 25:12 (1984), 3433–3438 | DOI | MR

[13] A. Kundu, “Integrable hierarchy of higher nonlinear Schrödinger type equations”, SIGMA, 2 (2006), 078, 12 pp. | DOI | MR | Zbl

[14] X.-B. Wang, B. Han, “The Kundu-nonlinear Schrödinger equation: breathers, rogue waves and their dynamics”, J. Phys. Soc. Japan, 89:1 (2020), 014001, 5 pp. | DOI

[15] C. Zhang, C. Li, J. He, “Darboux transformation and rogue waves of the Kundu-nonlinear Schrödinger equation”, Math. Methods Appl. Sci., 38:11 (2015), 2411–2425 | DOI | MR

[16] X.-B. Wang, B. Han, “Inverse scattering transform of an extended nonlinear Schrödinger equation with nonzero boundary conditions and its multisoliton solutions”, J. Math. Anal. Appl., 487:1 (2020), 123968, 20 pp. | DOI | MR

[17] X.-W. Yan, “Riemann–Hilbert method and multi-soliton solutions of Kundu-nonlinear Schrödinger equation”, Nonlinear Dynam., 102:4 (2020), 2811–2819 | DOI

[18] M. J. Ablowitz, P. A. Clarkson, Solutions, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149, Cambridge Univ. Press, Cambridge, 1991 | DOI | MR

[19] V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, 5, Springer, Berlin, 1991 | DOI | MR

[20] R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155, Cambridge Univ. Press, Cambridge, 2004 | DOI | MR

[21] G. W. Bluman, S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, 81, Springer, New York, 1989 | DOI | MR

[22] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, “Method for solving the Korteweg–de Vries equation”, Phys. Rev. Lett., 19:19 (1967), 1095–1097 | DOI

[23] E. Noether, “Invariante Variationsprobleme”, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 2 (1918), 235–275 | Zbl

[24] C. J. Papchristou, B. K. Harrison, “A method for constructing a Lax pair for the Ernst equation”, Electron. J. Theor. Phys., 6:22 (2009), 29–40, \par http://www.ejtp.com/articles/ejtpv6i22p29.pdf

[25] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “Nonlinear-evolution equations of physical significance”, Phys. Rev. Lett., 31:2 (1973), 125–127 | DOI | MR | Zbl

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

[27] R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams”, Phys. Rev. Lett., 13:15 (1965), 479–481 | DOI

[28] V. E. Zakharov, “Ustoichivost periodicheskikh voln konechnoi amplitudy na poverkhnost glubokoi zhidkosti”, PMTF, 1968, 86-94 | DOI

[29] A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers”, Appl. Phys. Lett., 23:3 (1973), 142–144 | DOI

[30] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR | Zbl

[31] G. Biondini, G. Kovac̆ic̆, “Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions”, J. Math. Phys., 55:3 (2014), 031506, 22 pp. | DOI | MR

[32] X. Zhou, “The Riemann–Hilbert problem and inverse scattering”, SIAM J. Math. Anal., 20:4 (1989), 966–986 | DOI | MR

[33] M. Kashiwara, “The Riemann–Hilbert problem for holonomic systems”, Publ. Res. Inst. Math. Sci., 20:2 (1984), 319–365 | DOI | MR

[34] A. S. Fokas, V. E. Zakharov, “The dressing method and nonlocal Riemann–Hilbert problems”, J. Nonlinear Sci., 2:1 (1992), 109–134 | DOI | MR

[35] D.-S. Wang, D.-J. Zhang, J. Yang, “Integrable properties of the general coupled nonlinear Schrödinger equations”, J. Math. Phys., 51:2 (2010), 023510, 17 pp. | DOI | MR

[36] S.-F. Tian, “The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method”, Proc. Roy. Soc. London Ser. A, 472:2195 (2016), 20160588, 22 pp. | DOI | MR

[37] X. Geng, J. Wu, “Riemann–Hilbert approach and $N$-soliton solutions for a generalized Sasa–Satsuma equation”, Wave Motion, 60 (2016), 62–72 | DOI | MR

[38] S.-F. Tian, “Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method”, J. Phys. A: Math. Theor., 50:39 (2017), 395204, 32 pp. | DOI | MR

[39] S.-F. Tian, “Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method”, J. Differ. Equ., 262:1 (2017), 506–558 | DOI | MR

[40] W.-X. Ma, “Riemann–Hilbert problems and $N$-soliton solutions for a coupled mKdV system”, J. Geom. Phys., 132 (2018), 45–54 | DOI | MR

[41] Y. Zhang, J. Rao, Y. Cheng, J. He, “Riemann–Hilbert method for the Wadati–Konno–Ichikawa equation: $N$ simple poles and one higher-order pole”, Phys. D, 399 (2019), 173–185 | DOI | MR

[42] W. Peng, S. Tian, X. Wang, T. Zhang, Y. Fang, “Riemann–Hilbert method and multi-soliton solutions for three-component coupled nonlinear Schrödinger equations”, J. Geom. Phys., 146 (2019), 103508, 9 pp. | DOI | MR

[43] J.-J. Yang, S.-F. Tian, Z.-Q. Li, Riemann–Hilbert approach to the inhomogeneous fifth-order nonlinear Schrödinger equation with non-vanishing boundary conditions, arXiv: 2001.08597

[44] P. Zhao, E. Fan, “Finite gap integration of the derivative nonlinear Schrödinger equation: a Riemann–Hilbert method”, Phys. D, 402 (2020), 132213, 31 pp. | DOI | MR

[45] C. Zhang, C. Li, J. He, “Rogue waves of the Kundu-nonlinear Schrödinger equation”, Open. J. Appl. Sci., 3 (2013), 94–98 | DOI

[46] A. S. Fokas, “A unified transform method for solving linear and certain nonlinear PDEs”, Proc. Roy. Soc. London Ser. A, 453:1962 (1997), 1411–1443 | DOI | MR