[Paper Review] Tau function and Virasoro action for the nxn KdV hierarchy

Chuu-Lian Terng, Karen Uhlenbeck|arXiv (Cornell University)|Jun 19, 2014

Nonlinear Waves and Solitons1 references3 citations

TL;DR

This paper establishes a geometric framework for the $n\times n$ KdV hierarchy by proving that the second partial derivatives of $\ln\tau_f$ reconstruct the formal inverse scattering solution $u_f$, and that the natural Virasoro action on $\ln\tau_f$ is realized via partial differential operators. It further proves a bijection between the $n\times n$ KdV hierarchy and the Gelfand-Dickey (GD$_n$) hierarchy, showing that the Virasoro actions on both hierarchies correspond under this equivalence.

ABSTRACT

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebra of L_+. Given f in L_-, there is a formal inverse scattering solution u_f of the hierarchy. When there is a 2 co-cycle that vanishes on both subalgebras of L_+ and L_-, Wilson constructed for each f in L_- a tau function tau_f for the hierarchy. In this third paper, we prove the following results for the nxn KdV hierarchy: (1) The second partials of ln(tau_f) are differential polynomials of the formal inverse scattering solution u_f. Moreover, u_f can be recovered from the second partials of ln(tau_f). (2) The natural Virasoro action on ln(tau_f) constructed in the second paper is given by partial differential operators in ln(tau_f). (3) There is a bijection between phase spaces of the nxn KdV hierarchy and the Gelfand-Dickey (GD_n) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection. (4) Our Virasoro action on the nxn KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GD_n hierarchy under the bijection.

Motivation & Objective

• To establish a uniform geometric framework for integrable systems, specifically the $n\times n$ KdV hierarchy.
• To prove that the formal inverse scattering solution $u_f$ can be reconstructed from the second partial derivatives of $\ln\tau_f$.
• To show that the natural Virasoro action on $\ln\tau_f$ is realized as a system of partial differential operators.
• To establish a bijection between the phase space of the $n\times n$ KdV hierarchy and the space of order $n$ linear differential operators, linking it to the Gelfand-Dickey (GD$_n$) hierarchy.
• To prove that the Virasoro action on the $n\times n$ KdV hierarchy corresponds to the known Virasoro action on the GD$_n$ hierarchy under this bijection.

Proposed method

• Utilizes a splitting of the loop group $L(SL(n,\mathbb{C}))$ into positive and negative subgroups $L_+$ and $L_-$, with associated Lie algebras $\mathcal{L}_+$ and $\mathcal{L}_-$.
• Applies Wilson's construction of the tau function $\tau_f$ for $f \in L_-$ using a 2-cocycle that vanishes on both $\mathcal{L}_+$ and $\mathcal{L}_-$.
• Employs gauge equivalence and Drinfeld-Sokolov quotient flows to relate the $n\times n$ KdV hierarchy to the Gelfand-Dickey hierarchy.
• Constructs cross sections of the gauge action and defines cross section flows to analyze the hierarchy structure.
• Uses reduced frames and the action of the Virasoro algebra on the reduced frame $M$ to derive the Virasoro vector fields on $\ln\tau_f$.
• Establishes the correspondence between the $n\times n$ KdV and GD$_n$ hierarchies via a bijection $\hat{\Psi}$, and proves that the induced Virasoro actions agree under this map.

Experimental results

Research questions

• RQ1 Can the formal inverse scattering solution $u_f$ be reconstructed from the second partial derivatives of $\ln\tau_f$?
• RQ2 Is the natural Virasoro action on $\ln\tau_f$ expressible as a system of partial differential operators?
• RQ3 Does a bijection exist between the phase space of the $n\times n$ KdV hierarchy and the space of order $n$ linear differential operators, such that the flows correspond?
• RQ4 Does the Virasoro action on the $n\times n$ KdV hierarchy correspond to the known Virasoro action on the GD$_n$ hierarchy under this bijection?
• RQ5 How is the Virasoro action on the $n\times n$ KdV hierarchy derived from the action on the negative group $L_-$?

Key findings

• The second partial derivatives of $\ln\tau_f$ are differential polynomials in the formal inverse scattering solution $u_f$, and $u_f$ can be fully recovered from these derivatives.
• The natural Virasoro action on $\ln\tau_f$ is realized as a system of partial differential operators acting on $\ln\tau_f$, confirming its geometric realizability.
• There exists a canonical bijection between the phase space of the $n\times n$ KdV hierarchy and the space of order $n$ linear differential operators, under which the flows of the two hierarchies correspond.
• The $n\times n$ KdV hierarchy is gauge-equivalent to the Gelfand-Dickey (GD$_n$) hierarchy, and solutions of one can be algorithmically transformed into solutions of the other.
• The Virasoro action on the $n\times n$ KdV hierarchy, constructed from the negative group action, corresponds exactly to the well-known Virasoro action on the GD$_n$ hierarchy under the bijection $\hat{\Psi}$.
• The Virasoro vector fields on the reduced frame $M$ are derived from the action of $\hat{\Gamma}$, and the resulting expressions match the known form of the GD$_n$ hierarchy's Virasoro action.