Discrete Toda lattices and the Laplace method
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We apply the Laplace cascade method to systems of discrete equations of the form u i+1,j+1 = f(u i+1,j , u i,j+1, u i,j , u i,j−1), where u ij , i, j ∈ ℤ, is an element of a sequence of unknown vectors. We introduce the concept of a generalized Laplace invariant and the related property that the systems is “of the Liouville type.” We prove a series of statements about the correctness of the definition of the generalized invariant and its applicability for seeking solutions and integrals of the system. We give some examples of systems of the Liouville type.
Keywordsnonlinear discrete equation Laplace method Darboux integrability
- 1.G. Darboux, Leçons sur la théorie des surfaces, Vol. 2, Gauthier-Villars, Paris (1889).Google Scholar
- 4.A. V. Zhiber, and V. V. Sokolov, Method of Laplace Cascade Integration and Darboux Integrable Equations [in Russian], RITs Bashkir State University, Ufa (1996).Google Scholar
- 7.A. M. Gurieva, “Method of Laplace cascade integration and nonlinear hyperbolic systems of equations [in Russian],” Doctoral dissertation, Ufa State Aviation University, Ufa (2005).Google Scholar
- 15.A. V. Zhiber, V. V. Sokolov, and S. Ya. Startsev, “Nonlinear hyperbolic systems of Liouville type [in Russian],” in: Materials of Intl. Conf. “Tikhonov-100”, Vol. 1 (2006), pp. 1–2.Google Scholar