Bell-polynomial approach and Wronskian determinant solutions for three sets of differential–difference nonlinear evolution equations with symbolic computation

  • Bo Qin
  • Bo Tian
  • Yu-Feng Wang
  • Yu-Jia Shen
  • Ming Wang
Article
  • 44 Downloads

Abstract

Under investigation in this paper are the Belov–Chaltikian (BC), Leznov and Blaszak–Marciniak (BM) lattice equations, which are associated with the conformal field theory, UToda\((m_1,m_2)\) system and r-matrix, respectively. With symbolic computation, the Bell-polynomial approach is developed to directly bilinearize those three sets of differential–difference nonlinear evolution equations (NLEEs). This Bell-polynomial approach does not rely on any dependent variable transformation, which constitutes the key step and main difficulty of the Hirota bilinear method, and thus has the advantage in the bilinearization of the differential–difference NLEEs. Based on the bilinear forms obtained, the N-soliton solutions are constructed in terms of the \(N \times N\) Wronskian determinant. Graphic illustrations demonstrate that those solutions, more general than the existing results, permit some new properties, such as the solitonic propagation and interactions for the BC lattice equations, and the nonnegative dark solitons for the BM lattice equations.

Keywords

Binary Bell polynomials Differential–difference nonlinear evolution equations Solitonic propagation and interactions Symbolic computation 

Mathematics Subject Classification

35C08 35F20 35G20 

References

  1. 1.
    Valeo, E., Oberman, C., Perkins, F.W.: Saturation of the decay instability for comparable electronic and ion temperatures. Phys. Rev. Lett. 28, 340 (1972)CrossRefGoogle Scholar
  2. 2.
    Agrawal, G.P.: Nonlinear Fiber Optics. Academic, New York (2002)MATHGoogle Scholar
  3. 3.
    Pitaevskii, L.P., Stringari, S.: Bose–Einstein Condensation. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  4. 4.
    Gao, X.Y.: Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation. Appl. Math. Lett. 73, 143 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gao, X.Y.: Backlund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid mechanics. Ocean Eng. 96, 245 (2015)CrossRefGoogle Scholar
  6. 6.
    Huang, Q.M., Gao, Y.T.: Wronskian, Pfaffian and periodic wave solutions for a (2+1)-dimensional extended shallow water wave equation. Nonlinear Dyn. 89, 2855 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Huang, Q.M., Gao, Y.T., Hu, L.: Breather-to-soliton transition for a sixth-order nonlinear Schrodinger equation in an optical fiber. Appl. Math. Lett. 75, 135 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Su, J.J., Gao, Y.T.: Dark solitons for a (2+1)-dimensional coupled nonlinear Schrodinger system with time-dependent coefficients in an optical fiber. Superlattices Microstruct. 104, 498 (2017)CrossRefGoogle Scholar
  9. 9.
    Deng, G.F., Gao, Y.T.: Integrability, solitons, periodic and travelling waves of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles. Eur. Phys. J. Plus 132, 255 (2017)CrossRefGoogle Scholar
  10. 10.
    Jia, S.L., Gao, Y.T., Zhao, C., Lan, Z.Z., Feng, Y.J.: Solitons, breathers and rogue waves for a sixth-order variable-coefficient nonlinear Schrodinger equation in an ocean or optical fiber. Eur. Phys. J. Plus 132, 34 (2017)Google Scholar
  11. 11.
    Ghosh, S., Sarma, D.: Bilinearization of \(N = 1\) supersymmetric modified KdV equations. Nonlinearity 16, 411 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Joshi, N., Lafortune, S., Ramani, A.: Hirota bilinear formalism and ultra-discrete singularity analysis. Nonlinearity 22, 871 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hone, A.N.W.: Lattice equations and \(\tau \)-functions for a coupled Painlevé system. Nonlinearity 15, 735 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Sakai, H.: Casorati determinant solutions for the \(q\)-difference sixth Painlevé equation. Nonlinearity 11, 823 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Anderson, B.P., Kasevich, M.A.: Macroscopic quantum interference from atomic tunnel arrays. Science 282, 1686 (1998)CrossRefGoogle Scholar
  16. 16.
    Trombettoni, A., Smerzi, A.: Discrete solitons and breathers with dilute Bose–Einstein condensates. Phys. Rev. Lett. 86, 2353 (2001)CrossRefGoogle Scholar
  17. 17.
    Kominis, Y., Bountis, T., Hizanidis, K.: Breathers in a nonautonomous Toda lattice with pulsating coupling. Phys. Rev. E 81, 066601 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zakharov, V.E., Musher, S.L., Rubenchik, A.M.: Nonlinear stage of parametric wave excitation in a plasma. JETP Lett. 19, 151 (1974)Google Scholar
  19. 19.
    Picard, G., Johnston, T.W.: Instability cascades, Lotka–Volterra population equations, and Hamiltonian chaos. Phys. Rev. Lett. 48, 1610 (1982)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Musher, S.L., Rubenchik, A.M., Zakharov, V.E.: Weak Langmuir turbulence. Phys. Rep. 252, 177 (1995)CrossRefGoogle Scholar
  21. 21.
    Marquié, P., Bilbault, J.M., Remoissenet, M.: Observation of nonlinear localized modes in an electrical lattice. Phys. Rev. E 51, 6127 (1995)CrossRefGoogle Scholar
  22. 22.
    Gerdjikov, V.S., Baizakov, B.B., Salerno, M., Kostov, N.A.: Adiabatic \(N\)-soliton interactions of Bose–Einstein condensates in external potentials. Phys. Rev. E 73, 046606 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Levi, D., Martina, L., Winternitz, P.: Lie-point symmetries of the discrete Liouville equation. J. Phys. A 48, 025204 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Baudouin, L., Ervedoza, S., Osses, A.: Stability of an inverse problem for the discrete wave equation and convergence results. J. Math. Pure Appl. 103, 1475 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Al-Ghassani, A., Halburd, R.G.: Height growth of solutions and a discrete Painlevé equation. Nonlinearity 28, 2379 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Yu, F.J.: Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz–Ladik equation. Phys. Rev. E 91, 032914 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  28. 28.
    Jia, T.T., Chai, Y.Z., Hao, H.Q.: Multi-soliton solutions and Breathers for the coupled nonlinear Schrodinger equations via the Hirota method. Math. Probl. Eng. 2016, 1741245 (2016)Google Scholar
  29. 29.
    Jia, T.T., Chai, Y.Z., Hao, H.Q.: Multi-soliton solutions and Breathers for the generalized coupled nonlinear Hirota equations via the Hirota method. Superlattices Microstruct. 105, 172 (2017)CrossRefGoogle Scholar
  30. 30.
    Deng, G.F., Gao, Y.T.: Solitons for the (3+1)-dimensional variable-coefficient coupled nonlinear Schrodinger equations in an optical fiber. Superlattices Microstruct. 109, 345 (2017)CrossRefGoogle Scholar
  31. 31.
    Huang, Q.M., Gao, Y.T., Jia, S.L., Wang, Y.L., Deng, G.F.: Bilinear Backlund transformation, soliton and periodic wave solutions for a (3+1)-dimensional variable-coefficient generalized shallow water wave equation. Nonlinear Dyn. 87, 2529 (2017)CrossRefGoogle Scholar
  32. 32.
    Su, J.J., Gao, Y.T.: Bilinear forms and solitons for a generalized sixth-order nonlinear Schrodinger equation in an optical fiber. Eur. Phys. J. Plus 132, 53 (2017)CrossRefGoogle Scholar
  33. 33.
    Weiss, J.: The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 24, 1405 (1983)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Estévez, P.G., Gordoa, P.R.: Darboux transformations via Painlevé analysis. Inverse Probl. 13, 939 (1997)CrossRefMATHGoogle Scholar
  35. 35.
    Alagesan, T., Chung, Y., Nakkeeran, K.: Painlevé test for the certain (2 \(+\) 1)-dimensional nonlinear evolution equations. Chaos Solitons Fract. 26, 1203 (2005)CrossRefMATHGoogle Scholar
  36. 36.
    Gilson, C., Lambert, F., Nimmo, J.J., Willox, R.: On the combinatorics of the Hirota \(D\)-operators. Proc. R. Soc. Lond. A 452, 223 (1996)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Lambert, F., Loris, I., Springael, J., Willox, R.: On a direct bilinearization method: Kaup’s higher-order water wave equation as a modified nonlocal Boussinesq equation. J. Phys. A 27, 5325 (1994)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lambert, F., Springael., J.: On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations. Chaos Solitons Fract. 12, 2821 (2001)CrossRefMATHGoogle Scholar
  39. 39.
    Lambert, F., Springael, J.: Soliton equations and simple combinatorics. Acta Appl. Math. 102, 147 (2008)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wang, Y.F., Tian, B., Wang, M.: Bell-Polynomial approach and integrability for the coupled Gross–Pitaevskii equations in Bose–Einstein condensates. Stud. Appl. Math. 131, 119 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Sun, W.R., Tian, B., Wang, Y.F., Zhen, H.L.: Soliton excitations and interactions for the three-coupled fourth-order nonlinear Schrödinger equations in the alpha helical proteins. Eur. Phys. J. D 69, 146 (2015)CrossRefGoogle Scholar
  42. 42.
    Qin, B., Tian, B., Liu, L.C., Wang, M., Lin, Z.Q., Liu, W.J.: Bell-polynomial approach and \(N\)-soliton solution for the extended Lotka–Volterra equation in plasmas. J. Math. Phys. 52, 043523 (2011)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Belov, A.A., Chaltikian, K.D.: Lattice analogues of \(W\)-algebras and classical integrable equations. Phys. Lett. B 309, 268 (1993)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Leznov, A.N.: Graded Lie algebras, representation theory, integrable mappings, and integrable systems. Theor. Math. Phys. 122, 211 (2000)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Blaszak, M., Marciniak, K.: \(R\)-matrix approach to lattice integrable systems. J. Math. Phys. 35, 4661 (1994)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Kupershmidt, B.A., Mathieu, P.: Quantum Korteweg–de Vries like equations and perturbed conformal field theories. Phys. Lett. B 227, 245 (1989)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Sasaki, R., Yamanaka, I.: Field theoretical construction of an infinite set of quantum commuting operators related with soliton equations. Commun. Math. Phys. 108, 691 (1987)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Hikami, K.: The Baxter equation for quantum discrete Boussinesq equation. Nucl. Phys. B 604, 580 (2001)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Hikami, K., Inoue, R.: Classical lattice \(W\) algebras and integrable systems. J. Phys. A 30, 6911 (1997)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Sahadevan, R., Khousalya, S.: Similarity reduction, generalized symmetries and integrability of Belov–Chaltikian and Blaszak–Marciniak lattice equations. J. Math. Phys. 42, 3854 (2001)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Toda, M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Jpn. 22, 431 (1967)CrossRefGoogle Scholar
  52. 52.
    Mokross, F., Büttner, H.: Comments on the diatomic Toda lattice. Phys. Rev. A 24, 2826 (1981)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Dreyer, W., Herrmann, M., Mielke, A.: Micro–macro transition in the atomic chain via Whitham’s modulation equation. Nonlinearity 19, 471 (2006)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Wang, H.Y., Hu, X.B., Tam, H.W.: On the two-dimensional Leznov lattice equation with self-consistent sources. J. Phys. A 40, 12691 (2007)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Lane, A.M., Thomas, R.G.: \(R\)-matrix theory of nuclear reactions. Rev. Mod. Phys. 30, 257 (1958)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Belhout, A., Ouichaoui, S., Beaumevieille, H., Boughrara, A., Fortier, S., Kiener, J., Maison, J.M., Mehdi, S.K., Rosier, L., Thibaud, J.P., Trabelsi, A., Vernotte, J.: Measurement and DWBA analysis of the \(12C (6Li, d)\) 16O \(\alpha \)-transfer reaction cross sections at \(48.2\) MeV. \(R\)-matrix analysis of \(12C (\alpha, \gamma ) 16O\) direct capture reaction data. Nucl. Phys. A 793, 178 (2007)CrossRefGoogle Scholar
  57. 57.
    Nemnes, G.A., Iona, L., Antohe, S.: Thermo-electrical properties of nanostructured ballistic nanowires in the \(R\)-matrix formalism using the Implicitly Restarted Arnoldi Method. Physica E 42, 1613 (2010)CrossRefGoogle Scholar
  58. 58.
    Sahadevan, R., Khousalya, S.: Master symmetries for Volterra equation, Belov–Chaltikian and Blaszak–Marciniak lattice equations. J. Math. Anal. Appl. 280, 241 (2003)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Zhang, D.J., Chen, D.Y.: The conservation laws of some discrete soliton systems. Chaos Solitons Fract. 14, 573 (2002)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Hu, X.B., Zhu, Z.N.: A Bäcklund transformation and nonlinear superposition formula for the Belov–Chaltikian lattice. J. Phys. A 31, 4755 (1998)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Hu, X.B., Tam, H.W.: Application of Hirota’s bilinear formalism to a two-dimensional lattice by Leznov. Phys. Lett. A 276, 65 (2000)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Hu, X.B., Zhu, Z.N.: Some new results on the Blaszak–Marciniak lattice: Bäcklund transformation and nonlinear superposition formula. J. Math. Phys. 39, 4766 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Bo Qin
    • 1
    • 2
  • Bo Tian
    • 1
  • Yu-Feng Wang
    • 1
    • 3
  • Yu-Jia Shen
    • 4
  • Ming Wang
    • 1
  1. 1.State Key Laboratory of Information Photonics and Optical Communications, and School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Department of Computer Science and EngineeringHong Kong University of Science and Technology Clear Water BayHong KongChina
  3. 3.College of ScienceMinzu University of ChinaBeijingChina
  4. 4.Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid DynamicsBeijing University of Aeronautics and AstronauticsBeijingChina

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