[Paper Review] Generalized KP Hierarchy for Several Variables

Francisco J. Plaza Martín|arXiv (Cornell University)|Aug 1, 2000

Polynomial and algebraic computation10 references3 citations

TL;DR

This paper generalizes the KP hierarchy to multiple variables using Sato's Grassmannian framework, establishing a 1-1 correspondence between wave functions and points in an infinite-dimensional Grassmannian. It introduces a method to construct finite gap algebraic solutions from algebro-geometric data, extending the Krichever map to higher dimensions.

ABSTRACT

Following the techniques of M. Sato (see \cite{Sa}), a generalization of the KP hierarchy for more than one variable is proposed. An approach to the classification of solutions and a method to construct algebraic solutions is also offered.

Motivation & Objective

• To extend the classical KP hierarchy to multiple variables using Sato's approach.
• To unify the classification of solutions and construction of algebraic solutions within a single framework.
• To generalize the Krichever map for higher-dimensional algebraic varieties.
• To provide a systematic method for generating finite gap solutions in the multivariable setting.
• To explore connections with algebro-geometric data such as divisors and local rings on N-dimensional schemes.

Proposed method

• Uses pseudodifferential operators in N variables with iterated Laurent series rings as coefficient fields.
• Defines the KP(N) hierarchy as a Lax system involving commuting operators $ L_i $ and their positive parts.
• Introduces a wave function $ \omega_U(t,x) $ as a formal oscillating function in time variables $ t_\alpha $ and spatial variables $ x $.
• Establishes a 1-1 correspondence between wave functions and points in the infinite Grassmannian $ \mathrm{Gr}(V) $, generalizing Sato's classification.
• Constructs solutions via the Krichever-type map from algebro-geometric data: a regular N-dimensional scheme $ X $, a point $ p $, local coordinates $ \alpha $, and ordered Weil divisors $ \{Y_1, \dots, Y_N\} $.
• Defines the solution space $ A_{\mathfrak{X}} $ as sections with non-negative valuation, showing it lies in $ \mathrm{Gr}(V) $ and yields a finite gap solution.

Experimental results

Research questions

• RQ1 How can the KP hierarchy be generalized to N variables while preserving the Sato-Grassmannian structure?
• RQ2 What is the precise correspondence between wave functions and points in the Grassmannian in the multivariable case?
• RQ3 Can finite gap solutions of the KP(N) hierarchy be systematically constructed from algebro-geometric data?
• RQ4 How does the Krichever map extend to higher-dimensional varieties in this framework?
• RQ5 What are the symmetries and central extensions of the group acting on the Grassmannian in the N > 1 case?

Key findings

• The KP(N) hierarchy is formulated as a Lax system with commuting operators $ L_i $ and compatibility conditions involving their positive parts.
• There exists a 1-1 correspondence between wave functions and points in the infinite Grassmannian $ \mathrm{Gr}(V) $, generalizing Sato's classification.
• The wave function $ \omega_U(t,x) $ satisfies the hierarchy as a compatibility condition of a system of differential equations.
• Finite gap solutions are constructed from algebro-geometric data $ \mathfrak{X} = (X,p,\alpha,\{Y_1,\dots,Y_N\}) $, where $ A_{\mathfrak{X}} $ is a subspace of sections with non-negative valuation.
• The solution space $ A_{\mathfrak{X}} $ lies in $ \mathrm{Gr}(V) $, and its wave function is a finite gap solution of the KP(N) hierarchy.
• The Lie algebra of automorphisms of the local ring $ \widehat{\mathcal{O}}_{X,p} \simeq \mathbb{C}[[x_1,\dots,x_N]] $ has no non-trivial central extensions for $ N > 1 $, differing from the $ N=1 $ case.