[Paper Review] n-th discrete KP hierarchy

Andrei K. Svinin|ArXiv.org|Jul 23, 2001

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper introduces the n-th discrete KP hierarchy as a generalization of the discrete KP hierarchy using pseudo-difference operators and a spectral parameter. It establishes an equivalence between the n-th discrete KP hierarchy and a bi-infinite sequence of differential KP hierarchies connected by two compatible gauge transformations, one of which is identified as a Darboux–Bäcklund transformation, thereby linking discrete and continuous integrable systems through τ-functions and wave operators.

ABSTRACT

We report an infinite class of discrete hierarchies which naturally generalize familiar discrete KP one.

Motivation & Objective

• To generalize the discrete KP hierarchy to an infinite class of n-th discrete KP hierarchies for arbitrary n ∈ ℕ.
• To establish a correspondence between the n-th discrete KP hierarchy and a bi-infinite sequence of differential KP hierarchies.
• To demonstrate that one of the gauge transformations linking the differential KP copies is a Darboux–Bäcklund transformation.
• To show that the wave function and τ-functions of the n-th discrete KP hierarchy satisfy bilinear identities analogous to those of the standard discrete KP.
• To provide a framework for understanding multi-matrix model solutions via discrete and differential integrable hierarchies.

Proposed method

• Introduces a pseudo-difference operator Q = Λ + a₀zⁿ⁻¹Λ¹⁻ⁿ + a₁z²⁽ⁿ⁻¹⁾Λ¹⁻²ⁿ + …, where Λ is the shift operator and aₖ depend on time variables t and spectral parameter z.
• Derives the Lax equations zᵖ⁽ⁿ⁻¹⁾∂ₚQ = [Q₊ᵖⁿ, Q] for p = 1,2,…, defining the n-th discrete KP hierarchy.
• Constructs wave functions Ψ(t,z) = Wχ(t,z) and dual wave functions Ψ*(t,z) = (W⁻¹)ᵀχ*(t,z), where W is a dressing operator in I + D₋.
• Establishes equivalence between the n-th discrete KP hierarchy and a sequence of differential KP operators 𝒬ᵢ = ŵᵢ∂ŵᵢ⁻¹ via two compatible gauge transformations: 𝒬ᵢ₊ₙ₋₁ = Gᵢ𝒬ᵢGᵢ⁻¹ and 𝒬ᵢ₊ₙ = Hᵢ𝒬ᵢHᵢ⁻¹.
• Derives evolution equations for Gᵢ and Hᵢ under KP flows: ∂ₚGᵢ = (𝒬ᵢ₊ₙ₋₁ᵖ)₊Gᵢ − Gᵢ(𝒬ᵢᵖ)₊ and similarly for Hᵢ.
• Identifies the transformation 𝒬ᵢ ↦ 𝒬ᵢ₊ₙ as a Darboux–Bäcklund transformation with eigenfunctions Φᵢ = τᵢ₊ₙ/τᵢ satisfying ∂ₚΦᵢ = (𝒬ᵢᵖ)₊Φᵢ.

Experimental results

Research questions

• RQ1 How can the discrete KP hierarchy be generalized to an infinite class of n-th discrete KP hierarchies for arbitrary n?
• RQ2 What is the precise relationship between the n-th discrete KP hierarchy and a sequence of differential KP hierarchies?
• RQ3 How do the gauge transformations Gᵢ and Hᵢ link the differential KP operators 𝒬ᵢ and what is their compatibility?
• RQ4 Can the n-th discrete KP hierarchy be interpreted as a chain of differential KP systems connected by Darboux–Bäcklund transformations?
• RQ5 What bilinear identities govern the τ-functions of the n-th discrete KP hierarchy, and how do they relate to the wave function structure?

Key findings

• The n-th discrete KP hierarchy is defined by the Lax equation zᵖ⁽ⁿ⁻¹⁾∂ₚQ = [Q₊ᵖⁿ, Q], which generates an infinite hierarchy of evolution equations for the coefficients aₖ(i) in the pseudo-difference operator Q.
• The wave function Ψ(t,z) satisfies the discrete linear system QΨ = zΨ and zᵖ⁽ⁿ⁻¹⁾∂ₚΨ = Q₊ᵖⁿΨ, with Ψ(t,z) = Wχ(t,z) and χ(t,z) = (zⁱe^{ξ(t,z)})ᵢ∈ℤ.
• The components ψᵢ(t,z) = z⁻ⁱΨᵢ(t,z) satisfy the differential-difference equations Gᵢψᵢ = zψᵢ₊ₙ₋₁ and Hᵢψᵢ = zψᵢ₊ₙ, where Gᵢ and Hᵢ are first-order differential operators.
• The Lax operators 𝒬ᵢ = ŵᵢ∂ŵᵢ⁻¹ are related by two compatible gauge transformations: 𝒬ᵢ₊ₙ₋₁ = Gᵢ𝒬ᵢGᵢ⁻¹ and 𝒬ᵢ₊ₙ = Hᵢ𝒬ᵢHᵢ⁻¹, with consistency condition Gᵢ₊ₙHᵢ = Hᵢ₊ₙ₋₁Gᵢ.
• The transformation 𝒬ᵢ ↦ 𝒬ᵢ₊ₙ is identified as a Darboux–Bäcklund transformation, with eigenfunctions Φᵢ = τᵢ₊ₙ/τᵢ satisfying ∂ₚΦᵢ = (𝒬ᵢᵖ)₊Φᵢ.
• The τ-functions of the n-th discrete KP hierarchy satisfy bilinear identities analogous to those of the standard discrete KP, with the key relation involving the residue and commutator structure of τ and its Darboux-transformed version.