[Paper Review] Formal solution to the KP hierarchy

S. M. Natanzon, A. Zabrodin|arXiv (Cornell University)|Sep 15, 2015

Nonlinear Waves and Solitons16 references3 citations

TL;DR

This paper provides a complete formal solution to the $α$-dependent KP hierarchy by constructing explicit series expansions for both the tau-function and its logarithm (the $F$-function) in terms of an infinite set of Cauchy-like initial data. The solution is given via a combinatorial recurrence for coefficients, enabling reconstruction of all formal solutions from arbitrary smooth or formal initial functions $f_0(x), f_1(x), \dots$, with explicit formulas for the $F$-function in terms of Schur functions and $α$-deformed derivatives.

ABSTRACT

We find all formal solutions to the $\hbar$-dependent KP hierarchy. They are characterized by certain Cauchy-like data. The solutions are found in the form of formal series for the tau-function of the hierarchy and for its logarithm (the F-function). An explicit combinatorial description of the coefficients of the series is provided.

Motivation & Objective

• To construct all formal solutions to the $α$-dependent KP hierarchy in the form of infinite formal power series.
• To address the lack of explicit series expansions for the $F$-function ($F = \hbar^2 \log \tau$) in the literature, despite such expansions being essential for applications in integrable systems and mathematical physics.
• To provide a complete characterization of solutions via an explicit recurrence procedure for coefficients, determined by an infinite set of initial data $f_0(x), f_1(x), \dots$, termed Cauchy-like data.
• To establish a rigorous algebraic framework using dual bases of symmetric functions and $α$-differential operators to describe the solution space.

Proposed method

• The solution is constructed by introducing $α$-deformed partial derivatives $\partial_i^\hbar$, which generalize standard derivatives and encode the $α$-dependence in the hierarchy.
• The $F$-function is expressed as a formal series in $\hbar$ using Schur functions $s_\lambda$ and structure constants derived from the inverse Kostka matrix $\kappa_\lambda = (L^{-1})_{\lambda(k)}$.
• The coefficients of the series are determined via a recursive procedure based on the Hirota bilinear equations for the $F$-function, formulated in terms of $\partial_i^\hbar$-operators.
• The method relies on a dual basis construction between monomial symmetric functions $m_\lambda$ and Schur functions $h_\lambda$, enabling explicit inversion of the series expansion.
• The solution is proven to be unique for any given set of Cauchy-like data $\{f_0(x), f_1(x), \dots\}$, with $f_k(x)$ related to the $k$-th $\hbar$-derivative of $F$ at $\mathbf{t}=0$ via a correction term of order $\hbar^{\ell(\lambda)-1}$.
• An inductive proof using Jacobi’s identity for matrix minors establishes the consistency of the solution with the Hirota bilinear identity for the tau-function.

Experimental results

Research questions

• RQ1 How can all formal solutions to the $\hbar$-KP hierarchy be systematically constructed in terms of initial data?
• RQ2 What is the explicit combinatorial structure of the coefficients in the formal series expansion of the $F$-function?
• RQ3 How do the $\hbar$-deformed derivatives $\partial_i^\hbar$ relate to the standard time derivatives in the KP hierarchy?
• RQ4 What is the precise relationship between the Cauchy-like data $f_k(x)$ and the $k$-th $\hbar$-derivative of the $F$-function at $\mathbf{t}=0$?
• RQ5 Can the $F$-function be reconstructed from an arbitrary set of initial functions $f_0(x), f_1(x), \dots$ via a closed-form recurrence?

Key findings

• All formal solutions to the $\hbar$-KP hierarchy are uniquely determined by an infinite set of Cauchy-like data $\{f_0(x), f_1(x), \dots\}$, where $f_0(x) = F(x; \mathbf{0})$.
• The $F$-function admits an explicit expansion $F(x; \mathbf{t}) = \sum_{k=1}^\infty \sum_{|\lambda|=k} \frac{\kappa_\lambda}{\rho(\lambda)} f_\lambda^\hbar(x) \hbar^{\ell(\lambda)-1}$, with $\kappa_\lambda = (L^{-1})_{\lambda(k)}$, providing a complete combinatorial description of the coefficients.
• The $\hbar$-deformed derivatives $\partial_i^\hbar$ are defined via $\partial_i^\hbar = \partial_{t_i} + \frac{\hbar}{2} \sum_{j=1}^\infty \frac{1}{j} \partial_{t_j} \partial_{t_{i+j}} + \cdots$, generalizing the standard time derivatives.
• The solution satisfies the Hirota bilinear identity for the tau-function, proven via induction and the Jacobi identity for minors of a matrix of $\partial_i^\hbar$-operators.
• The inverse relation $f_k^\hbar(x) = \partial_k F|_{\mathbf{t}=0} - k \sum_{|\lambda|=k, \ell(\lambda)>1} \frac{\kappa_\lambda}{\rho(\lambda)} f_\lambda^\hbar(x) \hbar^{\ell(\lambda)-1}$ allows reconstruction of the $\hbar$-deformed data from the $F$-function’s derivatives.
• The $\hbar \to 0$ limit of the solution yields the dispersionless KP hierarchy, and the formal series remain well-defined in this limit, even when smooth limits do not exist.