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Existence and Numerical Computation of Standing Wave Solutions for a System of Two Coupled Schrödinger Equations

  • Juan Carlos Muñoz GrajalesEmail author
  • Luisa Fernanda Vargas
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 275)

Abstract

In this paper, we consider the existence of a type of stationary wave of a system of two coupled Schrödinger equations with variable coefficients, which can be employed to describe the interaction among propagating modes in nonlinear optics and Bose-Einstein condensates (BECs), for instance. To prove existence of these solutions, we use some existing fixed point theorems for completely continuous operators defined in a cone in a Banach space. Furthermore, some numerical approximations of stationary waves are computed by using a spectral collocation technique combined with a Newton’s iteration.

Keywords

Schrödinger equations Numerical computations Standing waves 

Notes

Acknowledgments

This research was supported by Colciencias and Universidad del Valle, Calle 13 No. 100-00, Cali-Colombia, under the research project 1106-712-50006.

References

  1. 1.
    Muñoz Grajales, J.C.: Vector solitons of a coupled Schrödinger system with variable coefficients, Adv. Math. Phys. 2016, Article ID 5787508, 19 (2016)Google Scholar
  2. 2.
    Kivshar, YuS, Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003)Google Scholar
  3. 3.
    Sulem, C., Sulem, P.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, Berlin (1999)zbMATHGoogle Scholar
  4. 4.
    Malomed, B.A., Mihalache, D., Wise, F., Torner, L.: Spatio-temporal optical solitons. J. Opt. B Quantum Semiclassical Opt. 7, R53 (2005)CrossRefGoogle Scholar
  5. 5.
    Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85, 4502–5 (2000)CrossRefGoogle Scholar
  6. 6.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)CrossRefGoogle Scholar
  7. 7.
    Salerno, M., Konotop, V.V., Bludov,Yu.V.: Long-living Bloch oscillations of matter waves in periodic potentials. Phys. Rev. Lett. 101, 030405 (2008)Google Scholar
  8. 8.
    Ostrovskaya, E.A., Kivshar, Y.S., Lisak, M., Hall, B., Cattani, F.: Coupled-mode theory for Bose-Einstein condensates. Phys. Rev. A 61, 031601(R) (2000)CrossRefGoogle Scholar
  9. 9.
    Yang, Q., Zhang, J.F.: Bose-Einstein solitons in time-dependent linear potential. Opt. Commun. 258, 35–42 (2006)CrossRefGoogle Scholar
  10. 10.
    Radha, R., Vinayagam, P.S., Sudharsan, J.B., Malomed, B.A.: Persistent bright solitons in sign-indefinite coupled nonlinear Schrödinger equations with a time-dependent harmonic trap. Commun. Nonlinear Sci. Numer. Simul. 31(1–3), 30–39 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ieda, J., Miyakawa, T., Wadati, M.: Exact analysis of soliton dynamics in spinor Bose-Einstein condensates. Phys. Rev. Lett. 93, 194102 (2004)CrossRefGoogle Scholar
  12. 12.
    Feijoo, D., Paredes, A., Michinel, H.: Outcoupling vector solitons from a Bose-Einstein condensate with time-dependent interatomic forces. Phys. Rev. A 87, 063619 (2013)CrossRefGoogle Scholar
  13. 13.
    Yan, Z., Chow, K.W., Malomed, B.A.: Exact stationary wave patterns in three coupled nonlinear Schrödinger/Gross-Pitaevskii equations. Chaos, Solitons and Fractals 42, 3013–3019 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ho, T.L.: Spinor Bose condensates in optical traps. Phys. Rev. Lett. 81, 742 (1998)CrossRefGoogle Scholar
  15. 15.
    Law, C.K., Pu, H., Bigelow, N.P.: Quantum spins mixing in spinor Bose-Einstein condensates. Phys. Rev. Lett. 81, 5257 (1998)CrossRefGoogle Scholar
  16. 16.
    Ohmi, T., Machida, K.J.: Bose-Einstein condensation with internal degrees of freedom in alkali atom gases. J. Phys. Soc. Jpn. 67, 1822 (1998)CrossRefGoogle Scholar
  17. 17.
    Ieda, J., Miyakawa, T., Wadati, M.: Matter-wave solitons in an \(F=1\) spinor Bose-Einstein condensate. J. Phys. Soc. Jpn. 73, 2996 (2004)CrossRefGoogle Scholar
  18. 18.
    Wadati, M., Tsuchida, N.: Wave propagations in the \(F=1\) spinor Bose-Einstein condensates. J. Phys. Soc. Jpn. 75, 014301 (2006)CrossRefGoogle Scholar
  19. 19.
    Uchiyama, M., Ieda, J., Wadati, M.: Dark solitons in \(F=1\) spinor Bose-Einstein condensate. J. Phys. Soc. Jpn. 75, 064002 (2006)CrossRefGoogle Scholar
  20. 20.
    Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press (2001)Google Scholar
  21. 21.
    Belmonte-Beitia, J., Pérez-García, V.M., Brazhnyi, V.: Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. Commun. Nonlinear Sci. Numer. Simul. 16(1), 158–172 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Belmonte-Beitia, J., Pérez-García, V.M., Torres, P.J.: Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients. J. Nonlinear Sci. 19(4), 437–451 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Chu, J., O’Regan, D., Zhang, M.: Positive solutions and eigenvalue intervals for nonlinear systems. Proc. Indian Acad. Sci. (Math. Sci.) 117(1), 85–94 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jiang, D., Wei, J., Zhang, B.: Positive periodic solutions of functional differential equations and population models. Electron. J. Differ. Equ. 71(1–13) (2002)Google Scholar
  25. 25.
    Stuart, C.A.: Guidance properties of nonlinear planar waveguides. Arch. Ration. Mech. Anal. 125, 145–200 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Torres, P.J.: Guided waves in a multi-layered optical structure. Nonlinearity 19, 2103–2113 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zima, M.: On positive solutions of boundary value problems on the half-line. J. Math. Anal. Appl. 259, 127–136 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Jing Su, J., Gao, Y.: Dark solitons for a (2+1)-dimension coupled nonlinear Schrödinger system with time-dependent coefficients in an optical fiber. Superlattices Microstruct. 104, 498–508 (2017)CrossRefGoogle Scholar
  29. 29.
    Han, L., Huang, Y., Liu, H.: Solitons in coupled nonlinear Schrödinger equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3063–3073 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, L., Zhang, L., Zhu, Y., Qi, F., Wang, P., Guo, R., Li, M.: Modulational instability, nonautonomous characteristics and semirational solutions for the coupled nonlinear Schrödinger equations in inhomogeneous fibers. Commun. Nonlinear Sci. Numer. Simul. 40, 216–237 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Bose, S.N.: Plank’s law and light quantum hypothesis. Z. Phys. 26, 178–181 (1924)CrossRefGoogle Scholar
  32. 32.
    Anderson, M.H., Ensher, J.R., Mattews, M.R., Weiman, C.E., Cornell, E.A.: Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995)CrossRefGoogle Scholar
  33. 33.
    Theis, M., Thalhammer, G., Winkler, K., Hellwing, M., Ruff, G., Grimm, R., Hecker Denschlag, J.: Tuning the scattering length with an optically induced Feshbach resonance. Phys. Rev. Lett. 93, 123001 (2004)Google Scholar
  34. 34.
    Rodas-Verde, M.I., Michinel, H., Pérez-García, V.M.: Controllable soliton emission from a Bose-Einstein condensate. Phys. Rev. Lett. 95(15) 153903 (2005)Google Scholar
  35. 35.
    Teocharis, G., Schmelcher, P., Kevrekidis, P.G., Frantzeskakis, D.J.: Matter-wave solitons of collisionally inhomogeneous condensates. Phys. Rev. A 72, 033614 (2005)CrossRefGoogle Scholar
  36. 36.
    Primatorowa, M.T., Stoychev, K.T., Kamburova, R.S.: Interactions of solitons with extended nonlinear defects. Phys. Rev. E 72, 036608 (2005)CrossRefGoogle Scholar
  37. 37.
    Abdullaev, F.K., Garnier, J.: Propagation of matter-wave solitons in periodic and random nonlinear potentials. Phys. Rev. A 72, 061605 (2005)CrossRefGoogle Scholar
  38. 38.
    Garnier, J., Abdullaev, F.K.: Transmission of matter-wave solitons through nonlinear traps and barriers. Phys. Rev. A 74, 013604 (2006)CrossRefGoogle Scholar
  39. 39.
    Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. P. Noordhoff, Groningen (1964)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Juan Carlos Muñoz Grajales
    • 1
    Email author
  • Luisa Fernanda Vargas
    • 2
  1. 1.Departamento de MatemáticasUniversidad del ValleCaliColombia
  2. 2.Departamento de Ciencias Naturales y MatemáticasUniversidad JaverianaCaliColombia

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