Method for Generating Discrete Soliton Equations. I

We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

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“…is a discrete time Toda equation [3,16]. The (1, −P) periodic reduction of (44) corresponds to imposing the condition…”

Section: Lax Pair Associated With a Discrete Toda Equationmentioning

confidence: 99%

Some integrable maps and their Hirota bilinear forms

Hone

Kouloukas

Quispel

2017

J. Phys. A: Math. Theor.

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“…Indeed, it easy to check that the transformation 28) where f i are arbitrary functions, leaves the form of the equation unchanged. Equation ((2.25)) shows that the function Φ(a, s, u) is precisely of this form (the factor 2 asN is easily seen to be of this form, too).…”

Section: Normalization and Boundary Conditionsmentioning

confidence: 99%

Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics

2008

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We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K|M )-invariant R-matrix obey the Hirota bilinear difference equation. The nested Bethe ansatz for super spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation. Our basic tool is a chain of Bäcklund transformations for the Hirota equation connecting quantum transfer matrices. This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super spin chains.

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“…On the transcendental equation of ω a for a given k a in (5.5), Furthermore, we can construct other soliton solutions for p-adic equation (5.1) following the procedure in [6,7,8,11,25].…”

Section: P-adic Difference-difference Lotka-volterra Equationmentioning

confidence: 99%

“…Function form of finite type solution of (3.1) including soliton solution is completely determined at the infinity point of the spectral parameters k = ∞ [6,15]. The soliton solution is given by exponential functions whose power is polynomial of (k, s, t) owing to the algebraic properties of soliton solutions.…”

Section: It Implies That Classical Limit Might Be Regarded As Valuatimentioning

confidence: 99%

p‐adic difference‐difference Lotka‐Volterra equation and ultra‐discrete limit

Matsutani¹

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We study the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We point out that the structure of the space given by taking the ultra-discrete limit is the same as that of the p-adic valuation space. Since ultra-discrete limit can be regarded as a classical limit of a quantum object, it implies that a correspondence between classical and quantum objects might be associated with valuation theory.

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Method for Generating Discrete Soliton Equations. I

Cited by 212 publications

References 12 publications

Some integrable maps and their Hirota bilinear forms

Some integrable maps and their Hirota bilinear forms

Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics

p‐adic difference‐difference Lotka‐Volterra equation and ultra‐discrete limit

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