Riemann–Hilbert problem for the modified Landau–Lifshitz equation with nonzero boundary conditions

Abstract

We study a matrix Riemann–Hilbert (RH) problem for the modified Landau–Lifshitz (mLL) equation with nonzero boundary conditions at infinity. In contrast to the case of zero boundary conditions, multivalued functions arise during direct scattering. To formulate the RH problem, we introduce an affine transformation converting the Riemann surface into the complex plane. In the direct scattering problem, we study the analyticity, symmetries, and asymptotic behavior of Jost functions and the scattering matrix in detail. In addition, we find the discrete spectrum, residue conditions, trace formulas, and theta conditions in two cases: with simple poles and with second-order poles present in the spectrum. We solve the inverse problems using the RH problem formulated in terms of Jost functions and scattering coefficients. For further studying the structure of the soliton waves, we consider the dynamical behavior of soliton solutions for the mLL equation with reflectionless potentials. We graphically analyze some remarkable characteristics of these soliton solutions. Based on the analytic solutions, we discuss the influence of each parameter on the dynamics of the soliton waves and breather waves and propose a method for controlling such nonlinear phenomena.

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References

  1. 1

    Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, “Modification of the Landau–Lifshitz equation in the presence of a spin-polarized current in colossal- and giant-magnetoresistive materials,” Phys. Rev. B, 57, R3213–R23216 (1998); arXiv:cond-mat/9706132v1 (1997).

    ADS  Article  Google Scholar 

  2. 2

    J. C. Slonczewski, “Excitation of spin waves by an electric current,” J. Magn. Magn. Mater., 195, L261–L268 (1999).

    ADS  Article  Google Scholar 

  3. 3

    A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, “Magnetic solitons,” Phys. Rep., 194, 117–238 (1990).

    ADS  Article  Google Scholar 

  4. 4

    P.-B. He and W. M. Liu, “Nonlinear magnetization dynamics in a ferromagnetic nanowire with spin current,” Phys. Rev. B, 72, 064410 (2005).

    ADS  Article  Google Scholar 

  5. 5

    Z.-D. Li and Q.-Y. Li, “Dark soliton interaction of spinor Bose–Einstein condensates in an optical lattice,” Ann. Phys. (N. Y.), 322, 1961–1971 (2007); arXiv:1012.5469v1 [cond-mat.other] (2010).

    ADS  MATH  Article  Google Scholar 

  6. 6

    R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys., 14, 805–809 (1973).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  7. 7

    C.-Q. Su, Y.-Y. Wang, X.-Q. Liu, and N. Qin, “Conservation laws, modulation instability, and rogue waves for the localized magnetization with spin torque,” Commun. Nonlinear Sci. Numer. Simul., 48, 236–245 (2017).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. 8

    V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer Ser. Nonlin. Dyn., Vol. 5), Springer, Berlin (1991).

    MATH  Book  Google Scholar 

  9. 9

    Z.-D. Li, Q.-Y. Li, L. Li, and W. M. Liu, “Soliton solution for the spin current in a ferromagnetic nanowire,” Phys. Rev. E, 76, 026605 (2007); arXiv:0708.3120v1 [cond-mat.other] (2007).

    ADS  Article  Google Scholar 

  10. 10

    F. Zhao, Z.-D. Li, Q.-Y. Li, L. Wen, G. Fu, and W. M. Liu, “Magnetic rogue wave in a perpendicular anisotropic ferromagnetic nanowire with spin-transfer torque,” Ann. Phys., 327, 2085–2095 (2012); arXiv:1108.3252v2 [cond-mat.mtrl-sci] (2011).

    ADS  MATH  Article  Google Scholar 

  11. 11

    V. S. Gerdzhikov and P. P. Kulish, “On the multicomponent nonlinear Schrödinger equation in the case of nonvanishing boundary conditions [in Russian],” Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, 131, 34–46 (1983).

    MATH  Google Scholar 

  12. 12

    B. Prinari, M. J. Ablowitz, and G. Biondini, “Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions,” J. Math. Phys., 47, 063508 (2006).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. 13

    M. J. Ablowitz, G. Biondini, and B. Prinari, “Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions,” Inverse Problems, 23, 1711–1758 (2007).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. 14

    X.-D. Luo, “Inverse scattering transform for the complex reverse space–time nonlocal modified Korteweg–de Vries equation with nonzero boundary conditions and constant phase shift,” Chaos, 29, 073118 (2019).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. 15

    B. Prinari, G. Biondini, and A. D. Trubatch, “Inverse scattering transform for the multi-component nonlinear Schrödinger equation with nonzero boundary conditions,” Stud. Appl. Math., 126, 245–302 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  16. 16

    F. Demontis, B. Prinari, C. van der Mee, and F. Vitales, “The inverse scattering transform for the defocusing nonlinear Schrödinger equations with nonzero boundary condition,” Stud. Appl. Math., 131, 1–40 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  17. 17

    F. Demontis, B. Prinari, C. van der Mee, and F. Vitale, “The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions,” J. Math. Phys., 55, 101505 (2014).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. 18

    G. Biondini, E. Fagerstrom, and B. Prinari, “Inverse scattering transform for the defocusing nonlinear Schrödinger equation with fully asymmetric non-zero boundary conditions,” Phys. D, 333, 117–136 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  19. 19

    J. Ieda, M. Uchiyama, and M. Wadati, “Inverse scattering method for square matrix nonlinear Schrödinger equation under nonvanishing boundary conditions,” J. Math. Phys., 48, 013507 (2007); arXiv:nlin/0603010v2 (2006).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  20. 20

    G. Zhang and Z. Yan, “Inverse scattering transforms and \(N\)-double-pole solutions for the derivative NLS equation with zero/non-zero boundary conditions,” arXiv:1812.02387v1 [nlin.SI] (2018).

  21. 21

    J. Zhu and L. Wang, “Kuznetsov–Ma solution and Akhmediev breather for TD equation,” Commun. Nonlinear Sci. Numer. Simul., 67, 555–567 (2019).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. 22

    J. Zhu, L. Wang, and X. Geng, “Riemann–Hilbert approach to TD equation with nonzero boundary condition,” Front. Math. China., 13, 1245–1265 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  23. 23

    M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).

    MATH  Book  Google Scholar 

  24. 24

    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, 506–558 (2017).

    ADS  MATH  Article  Google Scholar 

  25. 25

    S.-F. Tian, “The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method,” Proc. Roy. Soc. London Ser. A, 472, 20160588 (2016).

    ADS  MATH  Google Scholar 

  26. 26

    W.-Q. Peng, S.-F. Tian, X.-B. Wang, T.-T. Zhang, and Y. Fang, “Riemann–Hilbert method and multi-soliton solutions for three-component coupled nonlinear Schrödinger equations,” J. Geom. Phys., 146, 103508 (2019).

    MathSciNet  MATH  Article  Google Scholar 

  27. 27

    J. J. Yang, S. F. Tian, W. Q. Peng, and T. T. Zhang, “The \(N\)-coupled higher-order nonlinear Schrödinger equation: Riemann–Hilbert problem and multi-soliton solutions,” Math. Meth. Appl. Sci., 43, 2458–2472 (2020).

    MATH  Article  Google Scholar 

  28. 28

    Z.-Q. Li, S.-F. Tian, W.-Q. Peng, and J.-J. Yang, “Inverse scattering transform and soliton classification of higher-order nonlinear Schrödinger–Maxwell–Bloch equations,” Theor. Math. Phys., 203, 709–725 (2020).

    MATH  Article  Google Scholar 

  29. 29

    D.-S. Wang, D.-J. Zhang, and J. Yang, “Integrable properties of the general coupled nonlinear Schrödinger equations,” J. Math. Phys., 51, 023510 (2010).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. 30

    L. Ai and J. Xu, “On a Riemann–Hilbert problem for the Fokas–Lenells equation,” Appl. Math. Lett., 87, 57–63 (2019).

    MathSciNet  MATH  Article  Google Scholar 

  31. 31

    Y. Zhang, Y. Cheng, and J. He, “Riemann–Hilbert method and \(N\)-soliton for two-component Gerdjikov–Ivanov equation,” J. Nonliner Math. Phys., 24, 210–223 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  32. 32

    B. Guo, N. Liu, and Y. Wang, “A Riemann–Hilbert approach for a new type coupled nonlinear Schrödinger equations,” J. Math. Anal. Appl., 459, 145–158 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  33. 33

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

    ADS  MATH  Article  Google Scholar 

  34. 34

    A. B. de Monvel and D. Shepelsky, “Riemann–Hilbert approach for the Camassa–Holm equation on the line,” C. R. Math. Acad. Sci. Paris, 343, 627–632 (2006).

    MathSciNet  MATH  Article  Google Scholar 

  35. 35

    D.-S. Wang, B. Guo, and X. Wang, “Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions,” J. Differ. Equ., 266, 5209–5253 (2019).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. 36

    S.-F. Tian and T.-T. Zhang, “Long-time asymptotic behavior for the Gerdjikov–Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition,” Proc. AMS, 146, 1713–1729 (2018).

    MATH  Article  Google Scholar 

  37. 37

    W.-X. Ma, “Riemann–Hilbert problems and \(N\)-soliton solutions for a coupled mKdV system,” J. Geom. Phys., 132, 45–54 (2018).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. 38

    X. Geng and J. Wu, “Riemann–Hilbert approach and \(N\)-soliton solutions for a generalized Sasa–Satsuma equation,” Wave Motion, 60, 62–72 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  39. 39

    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, 395204 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  40. 40

    A. A. Zabolotskii, “Solution of the reduced anisotropic Maxwell–Bloch equations by using the Riemann–Hilbert problem,” Phys. Rev. E, 75, 036612 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  41. 41

    L. A. Takhtajan and L. D. Faddeev, Hamiltonian Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987).

    Google Scholar 

  42. 42

    M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Univ. Press, Cambridge (2003).

    MATH  Book  Google Scholar 

  43. 43

    P. Henrici, Applied and Computational Complex Analysis, Wiley, New York (1974).

    MATH  Google Scholar 

  44. 44

    G. Biondini and G. Kovačič, “Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions,” J. Math. Phys., 55, 031506 (2014).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  45. 45

    N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover, New York (1986).

    MATH  Google Scholar 

  46. 46

    M. Pichler and G. Biondini, “On the focusing non-linear Schrödinger equation with non-zero boundary conditions and double poles,” IMA J. Appl. Math., 82, 131–151 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  47. 47

    G. Zhang, S. Chen, and Z. Yan, “Focusing and defocusing Hirota equations with non-zero boundary conditions: inverse scattering transforms and soliton solutions,” Commun. Nonlinear Sci. Numer. Simul., 80, 104927 (2020).

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgments

The authors thank the editor and a referee for their valuable comments and suggestions.

Funding

This research was supported by the National Natural Science Foundation of China (Grant No. 11975306), the Natural Science Foundation of Jiangsu Province (Grant No. BK20181351), the Six Talent Peaks Project in Jiangsu Province (Grant No. JY-059), the Fundamental Research Fund for the Central Universities (Grant Nos. 2019ZDPY07 and 2019QNA35), the Assistance Program for Future Outstanding Talents of China University of Mining and Technology (Grant No. 2020WLJCRCZL031), and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX20_2038).

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Correspondence to Shou-Fu Tian.

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Yang, JJ., Tian, SF. Riemann–Hilbert problem for the modified Landau–Lifshitz equation with nonzero boundary conditions. Theor Math Phys 205, 1611–1637 (2020). https://doi.org/10.1134/S0040577920120053

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Keywords

  • modified Landau–Lifshitz equation
  • matrix Riemann–Hilbert problem
  • nonzero boundary condition
  • soliton solution