Abstract

This paper is devoted to the study of the defocusing nonlinear Schrödinger equation with a self-consistent source and nonzero boundary conditions by the method of the inverse scattering problem. In cases where the source consists of a combination of eigenfunctions of the corresponding spectral problem for the Zakharov-Shabat system, the complete integrability of the nonlinear Schrödinger equation is investigated. Namely, the evolutions of the scattering data of the self-adjoint Zakharov-Shabat system, whose potential is a solution of the defocusing nonlinear Schrödinger equation with a self-consistent source, are obtained.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 3
Year: 2023

DOI: 10.12732/ijam.v36i3.4

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References

1. [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math., 53 (1974), 249–315.
2. [2] G. Biondini, E. Fagerstrom, B. Prinari, Inverse scattering transform for the defocusing nonlinear Schrödinger equation with fully asymmetric nonzero boundary conditions, Phys. D., 333 (2016), 117–136.
3. [3] G. Biondini , G. Kovačič, Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions, J. Math. Phys., 55 (2014), 031506, 22 pp.
4. [4] F. Demontis, B. Prinari, C. van der Mee, F. Vitale, The inverse scattering transform for the defocusing nonlinear Schrödinger equations with nonzero boundary conditions, Stud. Appl. Math., 131 (2013), 1–40.
5. [5] F. Demontis, B. Prinari, C. van der Mee, F. Vitale, The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions, J. Math. Phys., 55 (2014), 101505, 40.
6. [6] F. Demontis, C. van der Mee, Characterization of scattering data for the AKNS system, Acta Appl. Math., 131 (2014), 29–47.
7. [7] F. Demontis, C. van der Mee, A matrix Schrödinger approach to focusing nonlinear Schrödinger equations with nonvanishing boundary conditions, J. Nonlinear Sci., 32 (2022), 57, 29 pp.
8. [8] L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin (1987).
9. [9] A.B. Khasanov, A.A. Reı̆imberganov, On the finite-density solution of the nonlinear Schrödinger equation with a self-consistent source, Uzbek. Mat. Zh., (2009), 123–130.
10. [10] V.K. Melnikov, Integration of the nonlinear Schrödinger equation with a self-consistent source, Comm. Math. Phys., 137 (1991), 359–381.
11. [11] V.K. Melnikov, Integration of the nonlinear Schrödinger equation with a source, Inverse Problems, 8 (1992), 133–147.
12. [12] B. Prinari, F. Demontis, S. Li, T.P. Horikis, Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with nonzero boundary conditions, Phys. D, 368 (2018), 22–49.
13. [13] A.A. Reyimberganov, I.D. Rakhimov, The soliton solutions for the nonlinear Schrödinger equation with self-consistent source, Izv. Irkutsk. Gos. Univ. Ser. Mat., 36 (2021), 84–94.
14. [14] G.U. Urazboev, A.A. Reyimberganov, A.K. Babadjanova, Integration of the matrix nonlinear Schrödinger equation with a source, Izv. Irkutsk. Gos. Univ. Ser. Mat., 37 (2021), 63–76.
15. [15] A.B. Yakhshimuratov, The nonlinear Schrödinger equation with a selfconsistent source in the class of periodic functions, Math. Phys. Anal. Geom., 14 (2011), 153–169.
16. [16] V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Journal of Experimental and Theoretical Physics, 61 (1971), 118–134.
17. [17] V.E. Zakharov, A.B. Shabat, Interaction between solitons in a stable medium, Soviet Journal of Experimental and Theoretical Physics, 37 (1973), 823–828.