[Paper Review] The Atiyah-Hitchin bracket for the cubic nonlinear Schrodinger equation. ii. Periodic potentials

K. L. Vaninsky|arXiv (Cornell University)|Mar 16, 2004

Nonlinear Waves and Solitons12 references4 citations

TL;DR

This paper establishes a Poisson structure—called the deformed Atiyah–Hitchin bracket—on meromorphic functions over a hyperelliptic Riemann surface associated with periodic finite-gap potentials for the cubic nonlinear Schrödinger equation. By constructing a Weyl-type function 𝒳 on the spectral curve Γ, the authors show that the inverse spectral transform maps this abstract Poisson bracket to the classical Hamiltonian bracket on the physical phase space, providing a geometric realization of integrable dynamics via algebraic geometry.

ABSTRACT

This is the second in a series of papers on Poisson formalism for the cubic nonlinear Schrödinger equation with repulsive nonlinearity. In this paper we consider periodic potentials. The inverse spectral problem for the periodic auxiliary Dirac operator leads to a hyperelliptic Riemann surface $\G$. Using the spectral problem we introduce on this Riemann surface a meromorphic function $¶$. We call it the Weyl function, since it is closely related to the classical Weyl function discussed in the first paper. We show that the pair $(\G,¶)$ carries a natural Poisson structure. We call it the deformed Atiyah--Hitchin bracket. The Poisson bracket on the phase space is the image of the deformed Atiyah--Hitchin bracket under the inverse spectral transform.

Motivation & Objective

• To establish a geometric Poisson structure on the spectral data of periodic finite-gap solutions to the cubic nonlinear Schrödinger equation.
• To define a meromorphic Weyl function 𝒳 on a compact hyperelliptic Riemann surface Γ associated with the auxiliary Dirac operator.
• To introduce and characterize the deformed Atiyah–Hitchin bracket as a unified Poisson structure on meromorphic functions over Γ.
• To demonstrate that the physical space Poisson bracket arises as the image of this spectral Poisson bracket under the inverse spectral transform.

Proposed method

• Construct the spectral curve Γ as a hyperelliptic Riemann surface from the periodic inverse spectral problem of the Dirac operator.
• Define the Weyl function 𝒳(λ, w) as a meromorphic function on Γ, analytically continued across branch cuts.
• Introduce the deformed Atiyah–Hitchin bracket via a single formula valid globally on the compactified spectral curve, replacing piecewise definitions from prior work.
• Derive explicit Poisson bracket formulas for 𝒲(Q) and Ξ(P) using the functions Ω(Q), λ(Q), and Ξ(Q), with poles and symmetries encoded in the Riemann surface structure.
• Use canonical coordinates on Γ, such as λ(γₖ) and p(γₖ), to verify Poisson relations and connect to known integrable systems frameworks.
• Apply the inverse spectral transform to map the spectral Poisson bracket back to the physical phase space, recovering the classical Hamiltonian bracket.

Experimental results

Research questions

• RQ1 How can a global Poisson structure be defined on the spectral data of periodic finite-gap solutions to the cubic NLS equation?
• RQ2 What is the geometric meaning of the Weyl function 𝒳 on the compactified spectral curve Γ?
• RQ3 How does the deformed Atiyah–Hitchin bracket unify and generalize previous piecewise Poisson brackets on non-compact spectral covers?
• RQ4 Can the classical Hamiltonian bracket on the physical phase space be recovered as the image of a spectral Poisson bracket under the inverse spectral transform?
• RQ5 What role do the symmetries of the Riemann surface—such as ϵ± and ϵa actions—play in defining the Poisson structure?

Key findings

• The deformed Atiyah–Hitchin bracket is a globally defined Poisson structure on meromorphic functions over a compact hyperelliptic Riemann surface Γ, arising from the periodic inverse spectral problem of the cubic NLS equation.
• The Weyl function 𝒲(y, Q) is meromorphic on Γ, has 2(g+1) poles and zeros, and satisfies the symmetry 𝒲(y, ε±Q) = −𝒲(y, Q) and 𝒲(y, εaQ) = −𝒲(y, Q)*.
• The Poisson bracket between 𝒲(Q) and 𝒲(P) is given by a single rational formula involving Ξ(Q)−Ξ(P), Ω(Q), and Ω(P), with a denominator λ−μ.
• The bracket between Ξ(Q) and Ξ(P) is similarly expressed, showing consistency with the spectral curve’s algebraic structure.
• The canonical variables λ(γₖ) and p(γₖ) satisfy the standard Poisson relations: {λ(γₖ), λ(γₙ)} = 0, {p(γₖ), p(γₙ)} = 0, and {p(γₖ), λ(γₙ)} = δₖⁿ, confirming integrability.
• The inverse spectral transform maps the deformed Atiyah–Hitchin bracket to the classical Hamiltonian bracket on the physical phase space, establishing a complete geometric correspondence.