[Paper Review] Singular sector of the KP hierarchy, $\bar{\partial}$-operators of non-zero index and associated integrable systems

B. G. Konopelchenko, Luis Martı́nez Alonso+1 more|ArXiv.org|May 29, 1998

Nonlinear Waves and Solitons3 references3 citations

TL;DR

This paper investigates integrable hierarchies arising from the singular sector of the KP hierarchy via $ar{\partial}$-operators of non-zero index, using the $ar{\partial}$-dressing method to construct multidimensional equations with finite-codimension constraints. The key contribution is the identification of hidden KdV, Boussinesq, and Gelfand-Dikii hierarchies within this framework, extending the scope of integrable systems beyond standard KP structures.

ABSTRACT

Integrable hierarchies associated with the singular sector of the KP hierarchy, or equivalently, with $\dbar$-operators of non-zero index are studied. They arise as the restriction of the standard KP hierarchy to submanifols of finite codimension in the space of independent variables. For higher $\dbar$-index these hierarchies represent themselves families of multidimensional equations with multidimensional constraints. The $\dbar$-dressing method is used to construct these hierarchies. Hidden KdV, Boussinesq and hidden Gelfand-Dikii hierarchies are considered too.

Motivation & Objective

• To study integrable hierarchies associated with the singular sector of the KP hierarchy, particularly those linked to $ar{\partial}$-operators of non-zero index.
• To analyze how these hierarchies emerge as restrictions of the standard KP hierarchy to submanifolds of finite codimension in the space of independent variables.
• To extend the $ar{\partial}$-dressing method to construct families of multidimensional equations with multidimensional constraints for higher $ar{\partial}$-index.
• To identify and characterize hidden integrable systems such as KdV, Boussinesq, and Gelfand-Dikii hierarchies within this generalized framework.

Proposed method

• The $ar{\partial}$-dressing method is employed to generate solutions and hierarchies from $ar{\partial}$-operators of non-zero index.
• The construction involves restricting the standard KP hierarchy to submanifolds of finite codimension in the space of independent variables.
• The method systematically derives multidimensional equations with constraints by analyzing the kernel structure of $ar{\partial}$-operators.
• The approach uses complex analysis techniques, particularly the theory of $ar{\partial}$-problems, to derive bilinear identities and Lax equations.
• The framework allows for the derivation of both standard and hidden integrable hierarchies, including KdV and Boussinesq types.
• The paper establishes a correspondence between the index of the $ar{\partial}$-operator and the dimensionality and constraint structure of the resulting integrable systems.

Experimental results

Research questions

• RQ1 How do $ar{\partial}$-operators of non-zero index give rise to new integrable hierarchies within the KP framework?
• RQ2 What is the geometric and algebraic structure of the submanifolds of finite codimension that support these hierarchies?
• RQ3 How are hidden KdV and Boussinesq hierarchies embedded within the singular sector of the KP hierarchy via $ar{\partial}$-operators?
• RQ4 What constraints define the multidimensional equations derived from higher-index $ar{\partial}$-operators?
• RQ5 In what way does the $ar{\partial}$-dressing method generalize to accommodate non-zero index operators and yield new integrable systems?

Key findings

• The singular sector of the KP hierarchy, defined by $ar{\partial}$-operators of non-zero index, generates new integrable hierarchies distinct from the standard KP hierarchy.
• These hierarchies are realized as restrictions of the KP hierarchy to submanifolds of finite codimension in the space of independent variables.
• For higher $ar{\partial}$-index, the resulting systems are families of multidimensional equations with multidimensional constraints.
• The $ar{\partial}$-dressing method successfully constructs solutions and bilinear identities for these generalized hierarchies.
• Hidden KdV and Boussinesq hierarchies are identified as special cases within this framework, emerging from the singular sector.
• The Gelfand-Dikii hierarchy is also shown to be realizable through this approach, confirming the broad applicability of the method to integrable systems.