[Paper Review] hbar-Dependent KP hierarchy

Kanehisa Takasaki, Takashi Takebe|arXiv (Cornell University)|May 4, 2011

Nonlinear Waves and Solitons7 references3 citations

TL;DR

This paper presents a recursive construction of solutions to the ℏ-dependent KP hierarchy using ℏ-expansions of the dressing operator W = exp(X/ℏ) and the wave function Ψ = exp(S/ℏ). It establishes explicit recursion relations for the coefficients Xₙ and Sₙ, showing that the tau function admits an ℏ-expansion log τ = ℏ⁻²F₀ + ℏ⁻¹F₁ + F₂ + ⋯, with Fₙ determined recursively from the dispersionless limit solution F₀.

ABSTRACT

This is a summary of a recursive construction of solutions of the hbar-dependent KP hierarchy. We give recursion relations for the coefficients X_n of an hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). The asymptotic behaviours of (the logarithm of) the wave function and the tau function are also considered.

Motivation & Objective

• To develop a systematic method for constructing solutions to the ℏ-dependent KP hierarchy beyond the classical (dispersionless) limit.
• To establish a recursive procedure for computing the coefficients in the ℏ-expansion of the dressing operator W = exp(X/ℏ).
• To derive the corresponding ℏ-expansion of the wave function Ψ = exp(S/ℏ) and relate it to the dressing operator via total symbol calculus.
• To show that the tau function admits an ℏ-expansion log τ = ℏ⁻²F₀ + ℏ⁻¹F₁ + F₂ + ⋯, with Fₙ recursively determined from lower-order terms.
• To provide a framework connecting quantized canonical transformations to higher-order ℏ-corrections in integrable systems.

Proposed method

• Use of the exponential form W = exp(X/ℏ) with X = ∑ₙ₌₀^∞ ℏⁿXₙ to parametrize the dressing operator, enabling ℏ-expansion techniques.
• Application of Aoki's 'exponential calculus' for microdifferential operators to relate the total symbol of exp(X/ℏ) to the phase function S = ∑ₙ₌₀^∞ ℏⁿSₙ.
• Derivation of recursion relations for Xₙ from the Lax equations and the condition [f,g] = ℏ for canonical commuting operators f,g.
• Use of the wave function Ψ = W exp((xz + ζ(t,z))/ℏ) and its WKB form Ψ = exp(S/ℏ) to derive the ℏ-expansion of the tau function via logarithmic derivatives.
• Employment of the total symbol map σ_tot to translate operator-level relations into differential equations for the coefficients Sₙ and Xₙ.
• Derivation of the system ∂Fₙ/∂tⱼ = vₙ,ⱼ + ∑_{k+l=j} (1/l) ∂vₙ₋₁,ₗ/∂tₖ, with vₙ,ⱼ coefficients of Sₙ(z) = −∑ₖ z⁻ᵏ vₙ,ₖ / k, to recursively determine Fₙ.

Experimental results

Research questions

• RQ1 How can solutions of the ℏ-dependent KP hierarchy be systematically constructed beyond the dispersionless limit?
• RQ2 What recursion relations govern the ℏ-expansion coefficients Xₙ of the dressing operator W = exp(X/ℏ)?
• RQ3 How is the WKB form Ψ = exp(S/ℏ) of the wave function related to the coefficients Xₙ in the ℏ-expansion of X?
• RQ4 What is the structure of the ℏ-expansion of the tau function, and how are its coefficients Fₙ recursively determined?
• RQ5 How do the coefficients Sₙ of the phase function S relate to the coefficients Xₙ of the operator X?

Key findings

• The coefficients Xₙ in the ℏ-expansion of X are recursively determined from X₀, which corresponds to a solution of the dispersionless KP hierarchy.
• The wave function Ψ admits a WKB form Ψ = exp(S/ℏ), where S = ∑ₙ₌₀^∞ ℏⁿSₙ, and the coefficients Sₙ are recursively determined from the Xₙ via total symbol calculus.
• The phase function Sₙ is explicitly determined by X₀, ..., Xₙ through the exponential calculus of microdifferential operators.
• The tau function has an ℏ-expansion log τ = ℏ⁻²F₀ + ℏ⁻¹F₁ + F₂ + ⋯, with Fₙ recursively computable from the system ∂Fₙ/∂tⱼ = vₙ,ⱼ + ∑_{k+l=j} (1/l) ∂vₙ₋₁,ₗ/∂tₖ.
• The coefficients vₙ,ⱼ in the expansion Sₙ(z) = −∑ₖ z⁻ᵏ vₙ,ₖ / k are linked to the coefficients of the wave function's Laurent series and determine the Fₙ recursively.
• The framework establishes a one-to-one correspondence between solutions of the ℏ-dependent KP hierarchy and their ℏ-expansions, starting from the dispersionless limit.