B\"acklund transformations for Gelfand-Dickey flows, revisited

Abstract

We construct B\"acklund transformations (BT) for the Gelfand-Dickey hierarchy (GD$_n$-hierarchy) on the space of $n$-th order differential operators on the line. Suppose $L=\partial_x^n-\sum_{i=1}^{n-1}u_i\partial_x^{(i-1)}$ is a solution of the $j$-th GD$_n$ flow. We prove the following results: (1) There exists a system (BT)$_{u,k}$ of non-linear ordinary differential equations for $h:R^2\to C$ depending on $u_1, \ldots, u_{n-1}$ in $x$ and $t$ variables such that $\tilde L= (\partial+h)^{-1}L(\partial+h)$ is a solution of the $j$-th GD$_n$ flow if and only if $h$ is a solution of (BT)$_{u,k}$ for some parameter $k$. Moreover, coefficients of $\tilde L$ are differential polynomials of $u$ and $h$. We say such $\tilde L$ is obtained from a BT with parameter $k$ from $L$. (2) (BT)$_{u,k}$ is solvable. (3) There exists a compatible linear system for $\phi:R^2\to C$ depending on a parameter $k$, such that if $\phi_1, \ldots, \phi_{n-1}$ are linearly independent solutions of this linear system then $h:=(\ln W(\phi_1, \ldots, \phi_{n-1}))_x$ is a solution of (BT)$_{u,k}$ and $(\partial+h)^{-1} L (\partial+h)$ is a solution of the $j$-th GD$_n$ flow, where $W(\phi_1,\ldots,\phi_{n-1})$ is the Wronskian Moreover, these give all solutions of (BT)$_{u,k}$. (4) We show that the BT for the GD$_n$ hierarchy constructed by M. Adler is our BT with parameter $k=0$. (5) We construct a permutability formula for our BTs and infinitely many families of explicit rational solutions and soliton solutions.