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Register a new userIntegrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system. II
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Authors
Takayuki Tsuchida
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This is a continuation of our previous paper arXiv:1904.07924, which is devoted to the construction of integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system; in this series of papers, we refer to a discretization of one of the two spatial variables as a semi-discretization. In this paper, we construct an integrable semi-discrete Davey-Stewartson system, which is essentially different from the semi-discrete Davey-Stewartson system proposed in the previous paper arXiv:1904.07924. We first obtain integrable semi-discretizations of the two elementary flows that compose the Davey-Stewartson system by constructing their Lax-pair representations and show that these two elementary flows commute as in the continuous case. Then, we consider a linear combination of the two elementary flows to obtain a new integrable semi-discretization of the Davey-Stewartson system. Using a linear transformation of the continuous independent variables, one of the two elementary Davey-Stewartson flows can be identified with an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system proposed in https://link.aps.org/doi/10.1103/PhysRevE.91.062902 .
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The integrable Davey-Stewartson system is a linear combination of the two elementary flows that commute: $mathrm{i} q_{t_1} + q_{xx} + 2qpartial_y^{-1}partial_x (|q|^2) =0$ and $mathrm{i} q_{t_2} + q_{yy} + 2qpartial_x^{-1}partial_y (|q|^2) =0$. In the literature, each elementary Davey-Stewartson flow is often called the Fokas system because it was studied by Fokas in the early 1990s. In fact, the integrability of the Davey-Stewartson system dates back to the work of Ablowitz and Haberman in 1975; the elementary Davey-Stewartson flows, as well as another integrable $(2+1)$-dimensional nonlinear Schrodinger equation $mathrm{i} q_{t} + q_{xy} + 2 qpartial_y^{-1}partial_x (|q|^2) =0$ proposed by Calogero and Degasperis in 1976, appeared explicitly in Zakharovs article published in 1980. By applying a linear change of the independent variables, an elementary Davey-Stewartson flow can be identified with a $(2+1)$-dimensional generalization of the integrable long wave-short wave interaction model, called the Yajima-Oikawa system: $mathrm{i} q_{t} + q_{xx} + u q=0$, $u_t + c u_y = 2(|q|^2)_x$. In this paper, we propose a new integrable semi-discretization (discretization of one of the two spatial variables, say $x$) of the Davey-Stewartson system by constructing its Lax-pair representation; the two elementary flows in the semi-discrete case indeed commute. By applying a linear change of the continuous independent variables to an elementary flow, we also obtain an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system.
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.
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.
We consider a focusing Davey-Stewartson system and construct the solution of the Cauchy problem in the possible presence of exceptional points (and/or curves).
General high-order rogue wave solutions for the (1+1)-dimensional Yajima-Oikawa (YO) system are derived by using Hirotas bilinear method and the KP-hierarchy reduction technique. These rogue wave solutions are presented in terms of determinants in which the elements are algebraic expressions. The dynamics of first and higher-order rogue wave are investigated in details for different values of the free parameters. It is shown that the fundamental (first-order) rogue waves can be classified into three different patterns: bright, intermediate and dark ones. The high-order rogue waves correspond to the superposition of fundamental rogue waves. Especially, compared with the nonlinear Schodinger equation, there exists an essential parameter alpha to control the pattern of rogue wave for both first- and high-order rogue waves since the YO system does not possess the Galilean invariance.
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