Linearizing integral transform for the multicomponent lattice KP

Capel, H.W.; Wiersma, G.L.; Nijhoff, F.W.

doi:10.1016/0378-4371(86)90174-3

Cited by 20 publications

(11 citation statements)

References 32 publications

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“…Choosing the entry u = U 1,0 in the infinite matrix one can derive the KdV hierarchy and the first nontrivial equation is u x3 + 1 2 u x1x1x1 + 3 2 u 2 x1 = 0 which is the same as (4.22) up to some scaling on the independent variables and the dependent variable. The bilinear transform u = 2(ln τ ) x1 according to BKP gives us the bilinear form (D 4 1 + 2D 1 D 3 )τ · τ = 0 obviously. The infinite matrix structure for the linear variable u k in this case obeys…”

Section: Dimensional Reductionsmentioning

confidence: 99%

Linear integral equations, infinite matrices, and soliton hierarchies

Nijhoff

2018

Journal of Mathematical Physics

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A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all (2+1)-and (1+1)-dimensional soliton hierarchies associated with scalar differential spectral problems. The integrability characteristics for the obtained soliton hierarchies, including Miura-type transforms, τ -functions, Lax pairs as well as soliton solutions, are also derived within this framework.

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Section: Dimensional Reductionsmentioning

confidence: 99%

Linear integral equations, infinite matrices, and soliton hierarchies

Nijhoff

2018

Journal of Mathematical Physics

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“…The derivation of intermediate and full continuum limits of the lattice KP systems were studied in [22,23], as well as [7,30,31] where the derivation of hierarchy arising from the lattice KP equations was systematically studied. We will consider continuum limit on the level of the elliptic soliton solution.…”

Section: The Continuum Limitmentioning

confidence: 99%

Elliptic (N, N′)-soliton solutions of the lattice Kadomtsev-Petviashvili equation

Yoo-Kong

Nijhoff

2013

Journal of Mathematical Physics

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Elliptic soliton solutions, i.e., a hierarchy of functions based on an elliptic seed solution, are constructed using an elliptic Cauchy kernel, for integrable lattice equations of KadomtsevPetviashvili (KP) type. This comprises the lattice KP, modified KP (mKP) and Schwarzian KP (SKP) equations as well as Hirota's bilinear KP equation, and their successive continuum limits. The reduction to the elliptic soliton solutions of KdV type lattice equations is also discussed. The lattice KP and Hirota equationsThe study of the discrete versions of soliton systems, i.e., systems given by integrable partial difference equations, have become in recent years a focus of attention in the theory of integrable systems. Among those systems, the discrete analogue of Kadomtsev-Petviashvili (KP) equations which define in three dimensional lattice, seem to form a universal class of systems. The first equation of this type was found by Hirota in [12] and was referred to as DAGTE (Discrete analogue of generalised Toda equation) which is the bilinear equationwhere the Hirota operators D i produce finite forward-and backward shifts, when acting on a pair of functions, in the corresponding lattice direction, i.e.,Special reductions of this equation, are obtained when the coefficients a 1 , a 2 , a 3 satisfy the condition a 1 + a 2 + a 3 = 0, but the full equation is integrable in the sense of multidimensional consistency for arbitrary parameters, see [5]. Miwa [18] reparametrized the equation in that restricted case, and hence in that form it is often referred to as Hirota-Miwa equation 1 . In this paper we will investigate a class of solutions of this and related three-dimensional lattice equations, comprising the following equations:1 Sometimes the full equation (representing the blinear discrete KP equation) is also (in our view erroneously) referred to as the Hirota-Miwa equation. In fact, in [18] only the restricted case was considered, and generalized to a four-term equation which is nowadays referred to as the Miwa equation.

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“…Then the evolution of the matrix V may be presented in the form of the "Lax triad" 14) while there holds also the following identity: …”

Section: Lax Representationmentioning

confidence: 99%

“…A fruitful method is based on the "direct linearization" [44,51,14,82,45,43]. Its basic idea is to derive integrable nonlinear differential equations which are satisfied by the solutions of certain linear integral equations.…”

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confidence: 99%

Integrable Discretizations for Lattice System: Local Equations of Motion and Their Hamiltonian Properties

Suris

1999

Rev. Math. Phys.

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We develop the approach to the problem of integrable discretization based on the notion of r-matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non-locality. We demonstrate how to overcome this drawback. Namely, we introduce the notion of localizing changes of variables and construct such changes of variables for a large number of examples, including the Toda and the relativistic Toda lattices, the Volterra and the relativistic Volterra lattices, the second flows of the Toda and of the Volterra hierarchies, the modified Volterra lattice, the Belov-Chaltikian lattice, the Bogoyavlensky lattices, the Bruschi-Ragnisco lattice. We also introduce a novel class of constrained lattice KP systems, discretize all of them, and find the corresponding localizing change of variables. Pulling back the differential equations of motion under the localizing changes of variables, we find also (sometimes novel) integrable one-parameter deformations of integrable lattice systems. Poisson properties of the localizing changes of variables are also studied: they produce interesting one-parameter deformations of the known Poisson algebras.

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Linearizing integral transform for the multicomponent lattice KP

Cited by 20 publications

References 32 publications

Linear integral equations, infinite matrices, and soliton hierarchies

Linear integral equations, infinite matrices, and soliton hierarchies

Elliptic (N, N′)-soliton solutions of the lattice Kadomtsev-Petviashvili equation

Integrable Discretizations for Lattice System: Local Equations of Motion and Their Hamiltonian Properties

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