### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 071, 16 pages      arXiv:1704.07003      https://doi.org/10.3842/SIGMA.2017.071
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### $N$-Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation

Bao-Feng Feng a and Yasuhiro Ohta b
a) School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
b) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received April 25, 2017, in final form September 03, 2017; Published online September 06, 2017

Abstract
In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi-discrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component semi-discrete NLS equation. The asymptotic behavior is analysed for two-soliton solutions.

Key words: bright-dark soliton; semi-discrete vector NLS equation; Hirota's bilinear method; Pfaffian.

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References

1. Ablowitz M.J., Biondini G., Prinari B., Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions, Inverse Problems 23 (2007), 1711-1758.
2. Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge, 1991.
3. Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations, J. Math. Phys. 16 (1975), 598-603.
4. Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17 (1976), 1011-1018.
5. Ablowitz M.J., Ohta Y., Trubatch A.D., On discretizations of the vector nonlinear Schrödinger equation, Phys. Lett. A 253 (1999), 287-304, solv-int/9810014.
6. Ablowitz M.J., Prinari B., Trubatch A.D., Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Note Series, Vol. 302, Cambridge University Press, Cambridge, 2004.
7. Adler V.E., Postnikov V.V., On vector analogs of the modified Volterra lattice, J. Phys. A: Math. Gen. 41 (2008), 455203, 16 pages, arXiv:0808.0101.
8. Agrawal G.P., Nonlinear fiber optics, 5th ed., Elsevier Inc., London, 2013.
9. Barashenkov I.V., Getmanov B.S., Multisoliton solutions in the scheme for unified description of integrable relativistic massive fields. Non-degenerate ${\mathfrak{sl}}(2,{\mathbb C})$ case, Comm. Math. Phys. 112 (1987), 423-446.
10. Barashenkov I.V., Getmanov B.S., Kovtun V.E., Integrable model with nontrivial interaction between sub- and superluminal solitons, Phys. Lett. A 128 (1988), 182-186.
11. Barashenkov I.V., Getmanov B.S., Kovtun V.E., The unified approach to integrable relativistic equations: soliton solutions over nonvanishing backgrounds. I, J. Math. Phys. 34 (1993), 3039-3053.
12. Barashenkov I.V., Getmanov B.S., The unified approach to integrable relativistic equations: soliton solutions over nonvanishing backgrounds. II, J. Math. Phys. 34 (1993), 3054-3072.
13. Benney D.J., Newell A.C., The propagation of nonlinear wave envelopes, Stud. Appl. Math. 46 (1967), 133-139.
14. Dalfovo F., Giorgini S., Pitaevskii L.P., Stringari S., Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys. 71 (1999), 463-512, cond-mat/9806038.
15. Doliwa A., Santini P.M., Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy, J. Math. Phys. 36 (1995), 1259-1273, solv-int/9407005.
16. Dubrovin B.A., Malanyuk T.M., Krichever I.M., Makhan'kov V.G., Exact solutions of the time-dependent Schrödinger equation with self-consistent potentials, Sov. J. Part. Nucl. 19 (1988), 252-269.
17. Feng B.-F., General $N$-soliton solution to a vector nonlinear Schrödinger equation, J. Phys. A: Math. Theor. 47 (2014), 355203, 22 pages.
18. Hasegawa A., Kodama Y., Solitons in optical communications, Clarendon, Oxford, 1995.
19. Hasegawa A., Tappert F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomolous dispersion, Appl. Phys. Lett. 23 (1973), 142-144.
20. Hasegawa A., Tappert F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion, Appl. Phys. Lett. 23 (1973), 171-172.
21. Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge University Press, Cambridge, 2004.
22. Kanna T., Lakshmanan M., Tchofo Dinda P., Akhmediev N., Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations, Phys. Rev. E 73 (2006), 026604, 15 pages, nlin.SI/0511034.
23. Kivshar Y.S., Agrawal G.P., Optical solitons: from fibers to photonic crystals, Academic Press, San Diego, 2003.
24. Krökel D., Halas N.J., Giuliani G., Grischkowsky D., Dark-pulse propagation in optical fibers, Phys. Rev. Lett. 60 (1988), 29-32.
25. Makhan'kov V.G., Pashaev O.K., Nonlinear Schrödinger equation with noncompact isogroup, Theoret. and Math. Phys. 121 (1982), 979-987.
26. Manakov S.V., On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 38 (1974), 248-253.
27. Maruno K.-I., Ohta Y., Casorati determinant form of dark soliton solutions of the discrete nonlinear Schrödinger equation, J. Phys. Soc. Japan 75 (2006), 054002, 10 pages, nlin.SI/0506052.
28. Narita K., Soliton solution for discrete Hirota equation, J. Phys. Soc. Japan 59 (1990), 3528-3530.
29. Ohta Y., Pfaffian solution for coupled discrete nonlinear Schrödinger equation, Chaos Solitons Fractals 11 (2000), 91-95.
30. Ohta Y., Special solutions of discrete integrable systems, in Discrete Integrable Systems, Lecture Notes in Phys., Vol. 644, Editors B. Grammaticos, Y. Kosmann-Schwarzbach, T. Tamizhmani, Springer, Berlin, 2004, 57-83.
31. Ohta Y., Discretization of coupled nonlinear Schrödinger equations, Stud. Appl. Math. 122 (2009), 427-447.
32. Ohta Y., Hirota R., Tsujimoto S., Imai T., Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy, J. Phys. Soc. Japan 62 (1993), 1872-1886.
33. Ohta Y., Wang D.-S., Yang J., General $N$-dark-dark solitons in the coupled nonlinear Schrödinger equations, Stud. Appl. Math. 127 (2011), 345-371, arXiv:1011.2522.
34. Park Q.-H., Shin H.J., Systematic construction of multicomponent optical solitons, Phys. Rev. E 61 (2000), 3093-3106.
35. Prinari B., Ablowitz M.J., Biondini G., Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions, J. Math. Phys. 47 (2006), 063508, 33 pages.
36. Prinari B., Vitale F., Inverse scattering transform for the focusing Ablowitz-Ladik system with nonzero boundary conditions, Stud. Appl. Math. 137 (2016), 28-52.
37. Radhakrishnan R., Lakshmanan M., Bright and dark soliton solutions to coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen. 28 (1995), 2683-2692.
38. Sheppard A.P., Kivshar Y.S., Polarized dark solitons in isotropic Kerr media, Phys. Rev. E 55 (1997), 4773-4782.
39. Tsuchida T., Integrable discretization of coupled nonlinear Schrödinger equations, Rep. Math. Phys. 46 (2000), 269-278, nlin.SI/0002048.
40. Tsuchida T., Ujino H., Wadati M., Integrable semi-discretization of the coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen. 32 (1999), 2239-2262, solv-int/9903013.
41. Tsujimoto S., Integrable discretization of the integrable systems, in Applied Integrable Systems, Editor Y. Nakamura, Shokabo, Tokyo, 2000, 1-52.
42. Tsuzuki T., Nonlinear waves in the Pitaevskii-Gross equation, J. Low Temp. Phys. 4 (1971), 441-457.
43. van der Mee C., Inverse scattering transform for the discrete focusing nonlinear Schrödinger equation with nonvanishing boundary conditions, J. Nonlinear Math. Phys. 22 (2015), 233-264.
44. Vekslerchik V.E., Konotop V.V., Discrete nonlinear Schrödinger equation under nonvanishing boundary conditions, Inverse Problems 8 (1992), 889-909.
45. Vijayajayanthi M., Kanna T., Lakshmanan M., Bright-dark solitons and their collisions in mixed $N$-coupled nonlinear Schrödinger equations, Phys. Rev. A 77 (2008), 013820, 18 pages, arXiv:0711.4424.
46. Weiner A.M., Heritage J.P., Hawkins R.J., Thurston R.N., Kirschner E.M., Leaird D.E., Tomlinson W.J., Experimental observation of the fundamental dark soliton in optical fibers, Phys. Rev. Lett. 61 (1988), 2445-2448.
47. Yajima N., Oikawa M., A class of exactly solvable nonlinear evolution equations, Progr. Theoret. Phys. 54 (1975), 1576-1577.
48. Zakharov V.E., Collapse of Langumuir waves, Sov. Phys. JETP 35 (1972), 908-914.
49. Zakharov V.E., Shabat A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972), 62-69.
50. Zakharov V.E., Shabat A.B., Interaction betweem solitons in a stable medium, Sov. Phys. JETP 37 (1973), 823-828.
51. Zhang Y.J., Cheng Y., Solutions for the vector $k$-constrained KP hierarchy, J. Math. Phys. 35 (1994), 5869-5884.