Advertisement

A Darboux transformation for the Volterra lattice equation

  • Wen-Xiu Ma
Article
  • 13 Downloads

Abstract

A Darboux transformation is presented for the Volterra lattice equation, based on a pair of \(2\times 2\) matrix spectral problems. The resulting DT is applied to construction of solitary wave solutions from a constant seed solution. A particular phenomenon is that only one condition is required in determining the corresponding Darboux matrix, but not two as for most pairs of \(2\times 2 \) matrix spectral problems.

Keywords

Matrix spectral problem Integrable lattice equation Darboux transformation 

Mathematics Subject Classification

35Q51 35Q53 37K40 

Notes

Acknowledgements

The work was supported in part by NSFC under the Grants 11301454, 11301331, 11371086, 11571079 and 51771083, NSF under the Grant DMS-1664561, the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17KJB110020), Emphasis Foundation of Special Science Research on Subject Frontiers of CUMT under Grant No. 2017XKZD11, and the Distinguished Professorships by Shanghai University of Electric Power, China and North-West University, South Africa.

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interest.

References

  1. 1.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)CrossRefGoogle Scholar
  2. 2.
    Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Springer, Dordrecht (2005)CrossRefGoogle Scholar
  3. 3.
    Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21(5), 467–490 (1968)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)CrossRefGoogle Scholar
  5. 5.
    Novikov, S., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons—The Inverse Scattering Method. Consultants Bureau, New York (1984)zbMATHGoogle Scholar
  6. 6.
    Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  7. 7.
    Its, A.R.: “Isomonodromy solutions” of equations of zero curvature. Math. USSR Izv. 26(3), 497–529 (1986)CrossRefGoogle Scholar
  8. 8.
    Tu, G.Z.: On Liouville integrability of zero-curvature equations and the Yang hierarchy. J. Phys. A Math. Gen. 22(13), 2375–2392 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tu, G.Z.: A trace identity and its applications to the theory of discrete integrable systems. J. Phys. A: Math. Gen. 23(17), 3903–3922 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ma, W.X., Chen, M.: Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras. J. Phys. A Math. Gen. 39(34), 10787–10801 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ma, W.X.: A discrete variational identity on semi-direct sums of Lie algebras. J. Phys. A Math. Theor. 40(5), 15055–15069 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ma, W.X.: Darboux transformations for a Lax integrable system in \(2n\)-dimensions. Lett. Math. Phys. 39(1), 33–49 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ma, W.X., Zhang, Y.J.: Darboux transformatins of integrable couplings and applications. Rev. Math. Phys. 30(2), 1850003 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ma, W.X., Fuchssteiner, B.: Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations. J. Math. Phys. 40(5), 2400–2418 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Matveev, V.B., Salle, M.A.: Differential-difference evolution equations II: Darboux transformation for the Toda lattice. Lett. Math. Phys. 3(5), 425–429 (1979)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Geng, X.G.: Darboux transformation of the discrete Ablowitz–Ladik eigenvalue problem. Acta Math. Sci. 9(1), 21–26 (1989)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Xu, X.X., Yang, H.X., Sun, Y.P.: Darboux transformation of the modified Toda lattice equation. Mod. Phys. Lett. B 20(11), 641–648 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Xu, X.X.: A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and Darboux transformation. Appl. Math. Comput. 251, 275–283 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Wen, X.Y.: New hierarchies of integrable lattice equations and associated properties: Darboux transformation, conservation laws and integrable coupling. Rep. Math. Phys. 67(2), 259–277 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Volterra, V.: Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, Paris (1931)zbMATHGoogle Scholar
  21. 21.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, Y., Dong, H.H., Zhang, X.E., Yang, H.W.: Rational solutions and lump solutions to the generalized (3 + 1)-dimensional shallow water-like equation. Comput. Math. Appl. 73(2), 246–252 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Chen, S.T., Ma, W.X.: Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation. Front. Math. China 13(3), 525–534 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ma, W.X.: Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tang, Y.N., Tao, S.Q., Qing, G.: Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput. Math. Appl. 72(9), 2334–2342 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74(6), 1399–1405 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74(3), 591–596 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kofane, T.C., Fokou, M., Mohamadou, A., Yomba, E.: Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation. Eur. Phys. J. Plus 132, 465 (2017)CrossRefGoogle Scholar
  29. 29.
    Yang, J.Y., Ma, W.X., Qin, Z.Y.: Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation. Anal. Math. Phys. 8(3), 427–436 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ma, W.X., Yong, X.L., Zhang, H.Q.: Diversity of interaction solutions to the (2+1)-dimensional Ito equation. Comput. Math. Appl. 75(1), 289–295 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yang, J.Y., Ma, W.X., Qin, Z.Y.: Abundant mixed lump-soliton solutions to the BKP equation. East Asian J. Appl. Math. 8(2), 224–232 (2018)CrossRefGoogle Scholar
  32. 32.
    Dorizzi, B., Grammaticos, B., Ramani, A., Winternitz, P.: Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable? J. Math. Phys. 27(12), 2848–2852 (1986)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Konopelchenko, B., Strampp, W.: The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inverse Probl. 7(2), L17–L24 (1991)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Li, X.Y., Zhao, Q.L., Li, Y.X., Dong, H.H.: Binary Bargmann symmetry constraint associated with 3\(\times \)3 discrete matrix spectral problem. J. Nonlinear Sci. Appl. 8, 496–506 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhao, Q.L., Li, X.Y.: A Bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy. Anal. Math. Phys. 6(3), 237–254 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Dong, H.H., Zhang, Y., Zhang, X.E.: The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation. Commun. Nonlinear Sci. Numer. Simul. 36, 354–365 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Li, X.Y., Zhao, Q.L.: A new integrable symplectic map by the binary nonlinearization to the super AKNS system. J. Geom. Phys. 121, 123–137 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Geng, X.G., Wu, J.P.: Riemann–Hilbert approach and \(N\)-soliton solutions for a generalized Sasa–Satsuma equation. Wave Motion 60, 62–72 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Guo, B.L., Liu, N., Wang, Y.F.: A Riemann–Hilbert approach for a new type coupled nonlinear Schrdinger equations. J. Math. Anal. Appl. 459(1), 145–158 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of Mathematics and PhysicsShanghai University of Electric PowerShanghaiChina
  2. 2.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  4. 4.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  5. 5.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  6. 6.Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical ModellingNorth-West UniversityMmabathoSouth Africa

Personalised recommendations