Soliton solutions and Bäcklund transformation for the Kupershmidt five-field lattice: A bilinear approach. (English) Zbl 1009.37047

Summary: The Kupershmidt five-field lattice is considered. By a dependent variable transformation, the Kupershmidt lattice is transformed into a bilinear form by the introduction of three auxiliary variables. We present a Bäcklund transformation and a nonlinear superposition formula for the Kupershmidt lattice. As an application of the results, soliton solutions are derived.


37K60 Lattice dynamics; integrable lattice equations
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q51 Soliton equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
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[1] Kupershmidt, B.A., Discrete Lax equations and differential-difference calculus, Astérisque, 123, 212, (1985) · Zbl 0565.58024
[2] Blaszak, M.; Marciniak, K., r-matrix approach to lattice integrable systems, J. math. phys., 35, 4661-4682, (1994) · Zbl 0823.58013
[3] Hu, X.B.; Zhu, Z.N., Some new results on the blaszak-marciniak lattice: Bäcklund transformation and nonlinear superposition formula, J. math. phys., 39, 4766-4772, (1998) · Zbl 0927.37050
[4] Ma, W.X.; Hu, X.B.; Zhu, S.M.; Wu, Y.T., Bäcklund transformation and its superposition principle of a blaszak-marciniak four-field lattice, J. math. phys., 40, 6071-6086, (1999) · Zbl 1063.37564
[5] Hirota, R., Direct methods in soliton theory, () · Zbl 0124.21603
[6] Hirota, R.; Satsuma, J., A variety of nonlinear network equations generated from the Bäcklund transformation for the Toda lattice, Prog. theor. phys. suppl., 59, 64-100, (1976)
[7] Matsuno, Y., ()
[8] Nimmo, J.J.C., Hirota’s method, () · Zbl 0743.35074
[9] Hietarinta, J., A search for bilinear equations passing Hirota’s three-soliton condition, J. math. phys., 28, 1732-1742, (1987) · Zbl 0641.35073
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