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Register a new userThe Unified Soliton System as the ${rm AdS_2}$ System
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We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon equations.
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We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) $cong$ GL(2,$mathbb R$) $cong$ M{o}bius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the $N$-soliton solutions through the KdV type B{a}cklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the B{a}cklund transformations of the standard form into the same common Hirota form and the same common B{a}cklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the $N$-soliton solutions has same common form for KdV/mKdV/sinh-Gordon systems. Then the $N$-soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV $N$-soliton solutions. We also give general addition formulae coming from the KdV type B{a}cklund transformation which plays not only an important role to construct the trigonometric/hyperbolic $N$-soliton solutions but also an essential role to construct the elliptic $N$-soliton solutions. In contrast to the KdV type B{a}cklund transformation, the well-known mKdV/sinh-Gordon type B{a}cklund transformation gives the non-cyclic symmetric $N$-soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.
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.
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 .
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice for decaying initial data in the soliton region. In addition, we point out how to reduce the problem in the remaining region to the known case without solitons.
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