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تسجيل مستخدم جديدOn a new integrable generalization of the Toda lattice and a discrete Yajima-Oikawa system
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Takayuki Tsuchida
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We propose a new integrable generalization of the Toda lattice wherein the original Flaschka-Manakov variables are coupled to newly introduced dependent variables; the general case wherein the additional dependent variables are vector-valued is considered. This generalization admits a Lax pair based on an extension of the Jacobi operator, an infinite number of conservation laws and, in a special case, a simple Hamiltonian structure. In fact, the second flow of this generalized Toda hierarchy reduces to the usual Toda lattice when the additional dependent variables vanish; the first flow of the hierarchy reduces to a long wave-short wave interaction model, known as the Yajima-Oikawa system, in a suitable continuous limit. This integrable discretization of the Yajima-Oikawa system is essentially different from the discrete Yajima-Oikawa system proposed in arXiv:1509.06996 (also see https://link.aps.org/doi/10.1103/PhysRevE.91.062902) and studied in arXiv:1804.10224. Two integrable discretizations of the nonlinear Schrodinger hierarchy, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are contained in the generalized Toda hierarchy as special cases.
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A space discretization of an integrable long wave-short wave interaction model, called the Yajima-Oikawa system, was proposed in the recent paper arXiv:1509.06996 using the Hirota bilinear method (see also https://link.aps.org/doi/10.1103/PhysRevE.91
.062902). In this paper, we propose a Lax-pair representation for the discrete Yajima-Oikawa system as well as its multicomponent generalization also considered in arXiv:1509.06996 and prove that it has an infinite number of conservation laws. We also derive the next higher flow of the discrete Yajima-Oikawa hierarchy, which generalizes a modified version of the Volterra lattice. Relations to two integrable discrete nonlinear Schrodinger hierarchies, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are clarified.
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