×
Lu, Rong-Wu; Xu, Xi-Xiang; Zhang, Ning

Construction of solutions for an integrable differential-difference equation by Darboux-Bäcklund transformation. (English) Zbl 1428.34005

Appl. Math. Comput. 361, 389-397 (2019).
Summary: We first establish a one-fold Darboux-Bäcklund transformation for the integrable differential-difference equation which is presented in [the second author et al., Phys. Lett., A 349, No. 1–4, 153–163 (2006; Zbl 1195.37050)] by means of a proper gauge transformation matrix. Then, as a result of the \(N\) times one-fold Darboux-Bäcklund transformation, we derive \(N\)-fold Darboux-Bäcklund transformation. Finally, using the obtained Darboux-Bäcklund transformation, two exact solutions are worked out.

Cited in 3 Documents

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
35A30 Geometric theory, characteristics, transformations in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

integrable differential-difference equation; Lax pair; gauge transformation; Darboux-Bäcklund transformation; exact solution

Citations:

Zbl 1195.37050
Cite Review PDF
Full Text: DOI

References:

[1] Fermi, E.; Pasta, J.; Ulam, S., Collected Papers of Enrico Fermi II (1965), University of Chicago Press: University of Chicago Press Chicago
[2] Ablowitz, M.; Ladik, J., Nonlinear differential-difference equations, J. Math. Phys, 16, 598-603 (1975) · Zbl 0296.34062
[3] Toda, M., Theory of Nonlinear Lattices (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0694.70001
[4] Ruijsenaars, S. N., Relativistic Toda systems, Commun. Math. Phys., 133, 217-247 (1990) · Zbl 0719.58019
[5] Merola, I.; Ragnisco, O.; Tu, G. Z., A novel hierarchy of integrable lattice, Inverse Probl., 10, 1315-1334 (1994) · Zbl 0815.35105
[6] Xu, X. X.; Yang, H. X.; Ding, H. Y., A Liouville integrable lattice soliton equation,infinitely many conservation laws and integrable coupling systems, Phys. Lett., A349, 153-163 (2006) · Zbl 1195.37050
[7] Oevel, W.; Zhang, H. W.; Fuchssteiner, B., Mastersymmetries and multi-hamiltonian formulations for some integrable lattice systems, Prog. Theor. Phys., 81, 294-308 (1989)
[8] Tu, G. Z., A trace identity and its application to the theory of discrete integrable systems, J. Phys. A Math. Gen., 23, 3903-3922 (1990) · Zbl 0717.58027
[9] Ma, W. X.; Xu, X. X., Positive and negative hierarchies of integrable lattice models associated with a hamiltonian pair, Int. J. Theor. Phys., 43, 219-235 (2004) · Zbl 1058.37055
[10] Ma, W. X.; Xu, X. X., A modified Toda spectral problem and its hierarchy of bi-hamiltonian lattice equations, J. Phys. A Gen. Math., 37, 1323-1336 (2004) · Zbl 1075.37030
[11] Ma, W. X., A discrete variational identity on semi-direct sums of lie algebras, J.Phys. A Math.Theor., 40, 15055-15069 (2007) · Zbl 1128.22014
[12] Xu, X. X., Integrable couplings of relativistic Toda lattice systems in polynomial form and rational form, their hierarchies and bi-Hamiltonian structures, J. Phys. A Math. Theor., 42, 395201 (2009) · Zbl 1190.37077
[13] Matveev, V. B.; Salle, M. A., Darboux Transformations and Solitons (1991), Springer: Springer Berlin,Gemany · Zbl 0744.35045
[14] Gu, C. H.; Hu, H. S.; Zhou, Z. X., Soliton Theory and Its Application (1995), Zhejiang Science and Technology Publishing House: Zhejiang Science and Technology Publishing House Berlin, Germany · Zbl 0834.35003
[15] Wu, Y. T.; Geng, X. G., A new hierarchy integrable differential-difference equations and darboux transformation, J. Phys. A Math. Gen., 31, L677-L684 (1998) · Zbl 0931.35190
[16] Xu, X. X., Darboux transformation of a coupled lattice soliton equation, Phys. Lett.A., 349, 205-211 (2007) · Zbl 1197.37095
[17] 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) · Zbl 1328.37054
[18] Xu, X. X.; Xu, M., A family of integrable different-difference equations, its hamiltonian structure, and Darboux-Bäcklund transformation, Discrete. Dyn. Nat. Soc., 187, 11 (2018), . 4152917 · Zbl 1417.39062
[19] Ma, W. X., A Darboux transformation for the Volterra lattice equation, Anal. Math. Phys, 9, 1-8 (2019)
[20] Yu, J. P.; Ma, W. X.; Sun, Y. L.; Khalique, C. M., N-fold Darboux transformation and conservation laws of the modified Volterra lattice, Mod. Phys. Lett., 32, 1850409 (2018)
[21] Chen, S. T.; Ma, W. X., Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation, Front. Math. China., 13, 525-534 (2018) · Zbl 1403.35259
[22] Ma, W. X., Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs, J. Geom. Phys., 133, 45-54 (2018)
[23] 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, 427-436 (2018) · Zbl 1403.35261
[24] Ma, W. X.; Li, J.; Khalique, C. M., A study on lump solutions to a generalized Hirota-Satsuma-Ito equation in (2+1)-dimensions, Complexity, 2018, 9059858 (2018) · Zbl 1407.35177
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.