A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and Darboux transformation. (English) Zbl 1328.37054

Summary: A deformed reduced semi-discrete Kaup-Newell equation and its related integrable family are derived from discrete zero curvature equation. A Darboux transformation of Lax pair of this equation is established with the help of gauge transformation. By means of the resulting Darboux transformation, three exact solutions are given.


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
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Fermi, E.; Pasta, J.; Ulam, S., Collected papers of enrico Fermi II, (1965), University of Chicago Press Chicago
[2] Ablowitz, M.; Clarkson, P., Solitons, nonlinear evolution equations and inverse scattering, (1992), Cambridge Univ Press · Zbl 0762.35001
[3] Toda, M., Theory of nonlinear lattice, (1989), Springer-Verlag Berlin
[4] Flaschka, H., The Toda lattice, II: inverse scattering solution, Prog. Theor. Phys., 51, 703-716, (1974) · Zbl 0942.37505
[5] Zeng, Y. B.; Li, Y. S., New symplectic maps: integrability and Lax representation, Chin. Ann. Math., 18B, 457-466, (1997) · Zbl 0890.58018
[6] Pickering, A.; Zhu, Z. N., New integrable lattice hierarchies, Phys. Lett. A., 349, 439-445, (2006) · Zbl 1195.37040
[7] Oevel, W.; Zhang, H.; Fuchssteiner, B., Mastersymmetries and multi-Hamiltonian formulations for some integrable lattice systems, Prog. Theor. Phys., 81, 294-308, (1989)
[8] Fuchssteiner, B.; Ma, W. X., An approach to master symmetries of lattice equations, (Clarkson, P. A.; Nijhoff, F. W., Symmetries and Integrability of Difference Equations, (1999), Cambridge University Press Cambridge), 247 · Zbl 0923.58040
[9] Ma, W. X.; Fuchssteiner, B., Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations, J. Math. Phys., 40, 2400-2418, (1999) · Zbl 0984.37097
[10] Tu, G. Z., A trace identity and its applications to theory of discrete integrable systems, J. Phys. A: Math. Gen., 23, 3903-3922, (1990) · Zbl 0717.58027
[11] Blaszak, M.; Marciniak, K., R-matrix approach to lattice integrable systems, J. Math. Phys., 35, 4661-4682, (1994) · Zbl 0823.58013
[12] Zhang, D. Y.; Chen, D. Y., Hamiltonian structure of discrete soliton systems, J. Phys. A: Gen. Math., 35, 7225-7241, (2002) · Zbl 1039.37049
[13] Ma, W. X., A discrete variational identity on semi-direct sums of Lie algebras, J. Phys. A: Math. Theor., 40, 15055, (2007), 15pp · Zbl 1128.22014
[14] Xu, X. X., Integrable couplings of the relativistic Toda lattice systems in polynomial form and rational form, their hierarchies and bi-Hamiltonian structures, J. Phys. A: Math. Theor., 42, 395201, (2009), 21pp · Zbl 1190.37077
[15] 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
[16] Ma, W. X.; Xu, X. X.; Zhang, Y. F., Semidirect sums of Lie algebras and discrete integrable couplings, J. Math. Phys., 47, 053501, (2006), 16pp · Zbl 1111.37059
[17] Ma, W. X.; Geng, X. G., Bäcklund transformations of soliton systems from symmetry constraints, CRM Proc. Lect. Notes, 29, 313-323, (1999) · Zbl 1010.37048
[18] Xu, X. X.; Zhang, Y. F., A hierarchy of Lax integrable lattice equations, Liouville integrability and a new integrable symplectic map, Commun. Theor. Phys. (Beijing, China), 41, 321-328, (2004) · Zbl 1167.37354
[19] Ma, W. X.; Maruno, K., Complexiton solutions of the Toda lattice equation, Physica A, 343, 219-237, (2004)
[20] Wu, Y. T.; Geng, X. G., A new integrable symplectic map associated with lattice soliton equations, J. Math. Phys., 37, 2338-2345, (1996) · Zbl 0864.58028
[21] Xu, X. X., Darboux transformation of a coupled lattice soliton equation, Phys. Lett. A, 362, 205-211, (2007) · Zbl 1197.37095
[22] Yang, H. Y.; Xu, X. X.; Ding, H. Y., New hierarchies of integrable positive and negative lattice models and Darboux transformation, Chaos Solitons Fract., 26, 1091-1103, (2005) · Zbl 1081.37040
[23] Tsuchida, T., Integrable discretizations of derivative nonlinear schrodinger equations, J. Phys. A: Math. Gen., 35, 7827-7847, (2002) · Zbl 1040.37061
[24] Yan, Z. Y., Discrete exact solutions of modified Volterra and Volterra lattice equations via the new discrete sine-Gordon expansion algorithm, Chaos Solitons Fract., 64, 1798-1811, (2006) · Zbl 1113.37054
[25] Hu, X. B.; Wu, Y. T., A new integrable differential-difference system and its explicit solutions, J. Phys. A: Math. Gen., 32, 1515-1521, (1999) · Zbl 0930.35183
[26] Bekir, A., Application of the exp-function method for nonlinear differential-difference equations, Appl. Math. Comput., 215, 4049-4053, (2010) · Zbl 1185.35312
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