A Darboux transformation for the Volterra lattice equation

Analysis and Mathematical Physics - Tập 9 Số 4 - Trang 1711-1718 - 2019
Wen‐Xiu Ma1
1College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China

Tóm tắt

Từ khóa


Tài liệu tham khảo

Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)

Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Springer, Dordrecht (2005)

Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21(5), 467–490 (1968)

Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)

Novikov, S., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons—The Inverse Scattering Method. Consultants Bureau, New York (1984)

Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge University Press, Cambridge (2000)

Its, A.R.: “Isomonodromy solutions” of equations of zero curvature. Math. USSR Izv. 26(3), 497–529 (1986)

Tu, G.Z.: On Liouville integrability of zero-curvature equations and the Yang hierarchy. J. Phys. A Math. Gen. 22(13), 2375–2392 (1989)

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)

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)

Ma, W.X.: A discrete variational identity on semi-direct sums of Lie algebras. J. Phys. A Math. Theor. 40(5), 15055–15069 (2007)

Ma, W.X.: Darboux transformations for a Lax integrable system in $$2n$$ 2 n -dimensions. Lett. Math. Phys. 39(1), 33–49 (1997)

Ma, W.X., Zhang, Y.J.: Darboux transformatins of integrable couplings and applications. Rev. Math. Phys. 30(2), 1850003 (2018)

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)

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)

Geng, X.G.: Darboux transformation of the discrete Ablowitz–Ladik eigenvalue problem. Acta Math. Sci. 9(1), 21–26 (1989)

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)

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)

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)

Volterra, V.: Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, Paris (1931)

Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)

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)

Chen, S.T., Ma, W.X.: Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation. Front. Math. China 13(3), 525–534 (2018)

Ma, W.X.: Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)

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)

Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74(6), 1399–1405 (2017)

Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74(3), 591–596 (2017)

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)

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)

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)

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)

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)

Konopelchenko, B., Strampp, W.: The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inverse Probl. 7(2), L17–L24 (1991)

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)

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)

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)

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)

Geng, X.G., Wu, J.P.: Riemann–Hilbert approach and $$N$$ N -soliton solutions for a generalized Sasa–Satsuma equation. Wave Motion 60, 62–72 (2016)

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)