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Reflectionless Solutions for Square Matrix NLS with Vanishing Boundary Conditions

  • Francesco DemontisEmail author
  • Cornelis van der Mee
Article
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Abstract

In this article we derive the reflectionless solutions of the 2 + 2 matrix NLS equation with vanishing boundary conditions and four different symmetries by using the matrix triplet method of representing the Marchenko integral kernel in separated form. Apart from using the Marchenko method, these solutions are also verified by direct substitution in the 2 + 2 NLS equation.

Keywords

Matrix Nonlinear Schroedinger equation Inverse scattering transform Matrix triplets method Reflectionless solutions 

Mathematics Subject Classification (2010)

37K10 37K40 45Q05 

Notes

Acknowledgments

The research leading to this article has been partially supported by the Regione Autonoma della Sardegna in the framework of the research programs Integro-Differential equations and non-local problems and Algorithms and Models for Imaging Science [AMIS], and by INdAM-GNFM (Istituto Nazionale di Alta Matematica, National Institute of Advanced Mathematics – Gruppo Nazionale per la Fisica Matematica, National Group for Mathematical Physics).

References

  1. 1.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform – Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and continuous nonlinear Schrödinger systems. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  3. 3.
    Aktosun, T., Busse, T.h., Demontis, F., van der Mee, C.: Symmetries for exact solutions to the nonlinear Schrödinger equation. J. Phys. A 43, 025202 (2010)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Problems 23, 2171–2195 (2007)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal factorization of matrix and operator functions, Birkhäuser OT 1, Basel (1979)Google Scholar
  6. 6.
    Camassa, R., Falqui, G., Ortenzi, G., Pitton, G.: Singularity formation as a wetting formalism in a dispersionless water wave model. Nonlinearity 32, 4079–4116 (2019)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., Thomson, C.: Hydrodynamic models and confinement effects by horizontal boundaries. J. Nonlin. Sci. 29, 1445–1498 (2019)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Demontis, F.: Direct and inverse scattering of the matrix Zakharov-Shabat system, Ph.D. thesis, University of Cagliari, 2007; also: Lambert Acad. Publ., Saarbrücken (2012)Google Scholar
  9. 9.
    Demontis, F., Lombardo, S., Sommacal, M., Vargiu, F.: Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation. Commun. Nonlinear Sci. Numer. Simul. 64, 35–65 (2018)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Demontis, F., Ortenzi, G., van der Mee, C.: Exact Solutions of the Hirota Equation and Vortex Filaments Motion. Physica D 313, 61–80 (2015)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Demontis, F., Prinari, B., van der Mee, C., Vitale, F.: The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions. J. Math. Phys. 55(101505), 40 (2014)zbMATHGoogle Scholar
  12. 12.
    Demontis, F., van der Mee, C.: Marchenko equations and norming constants of the matrix Zakharov-Shabat system. Operators and Matrices 2, 79–113 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Demontis, F., van der Mee, C.: Explicit solutions of the cubic matrix nonlinear Schrödinger equation. Inverse Problems 24, 025020 (2008)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Eckert, K., Zawitkowski, Ł., Leskinen, M.J., Sanpera, A., Lewenstein, M.: Ultracold atomic Bose and Fermi spinor gases in optical lattices. New Journal of Physics 9(5), 133 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Faddeev, L.D., Takhtajan, L.D.: Hamiltonian methods in the theory of Solitons. Springer, Berlin and Heidelberg (2007)zbMATHGoogle Scholar
  16. 16.
    Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. John Hopkins University Press, Baltimore (2013)zbMATHGoogle Scholar
  17. 17.
    Grabowski, M.: Second harmonic generation in periodically modulated media, Photonic Band Gaps and Localization, NATO ASI Series B, Vol. 308, 453–458, Springer US, Boston (1993)Google Scholar
  18. 18.
    Grabowski, M.: Bichromatic wave propagation in periodically poled media. Phys. Rev. A 48, 2370–2373 (1993)ADSCrossRefGoogle Scholar
  19. 19.
    Hirota, R.: The direct method in Soliton theory. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  20. 20.
    Ho, T.-L., Yip, S.: Pairing of fermions with arbitrary spin. Phys. Rev. Lett. 82, 247–250 (1999)ADSCrossRefGoogle Scholar
  21. 21.
    Ieda, J., Miyakawa, T., Wadati, M.: Exact analysis of soliton dynamics in spinor Bose-einstein condensates. Phys. Rev. Lett. 93, 194102 (2004)ADSCrossRefGoogle Scholar
  22. 22.
    Ieda, J., Uchiyama, M., Wadati, M.: Dark soliton in F = 1 spinor Bose-Einstein condensates. J. Phys. Soc. Jpn 75, 064002 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    Lunquist, P.B., Andersen, D.R., Kivshar, Y.S.: Multicolor solitons due to four wave mixing. Phys. Rev. E 57, 3551–3555 (1998)ADSCrossRefGoogle Scholar
  24. 24.
    Moro, A., Trillo, S.: Mechanism of wave breaking from a vacuum point in the defocusing nonlinear Schrödinger equation. Phys. Rev. E 89, 023202 (2014)ADSCrossRefGoogle Scholar
  25. 25.
    Prinari, B., Demontis, F., Li, S., Horikis, T.: Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with non-zero boundary conditions. Physica D 368, 22–49 (2018)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Prinari, B., Ortiz, A., van der Mee, C., Grabowski, M.: Inverse scattering transform and solitons for square matrix nonlinear matrix nonlinear Schrödinger equations. Stud. Appl. Math. 141, 308–352 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Uchiyama, M., Ieda, J-i, Wadati, M.: Inverse scattering method for square matrix nonlinear Schroedinger equation under nonvanishing boundary conditions. J. Math. Phys. 48, 013507 (2007)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    van der Mee, C.: Nonlinear evolution models of integrable type, SIMAI e-Lecture Notes 11. Torino (2013)Google Scholar
  29. 29.
    Wu, C.: Competing orders in one dimensional spin-3/2 fermionic systems. Phys. Rev. Lett. 95, 266404 (2005)ADSCrossRefGoogle Scholar
  30. 30.
    Wu, C.: Hidden symmetry and quantum phases in spin-3/2 cold atomic systems. Modern Phys. Lett B 20, 1707–1738 (2006)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di CagliariCagliariItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità di CagliariCagliariItaly

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