Bäcklund transformations for Gelfand-Dickey flows, revisited

Abstract

We construct Bäcklund 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 $u_1,\dots ,u_{n-1}$ 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 $\varphi :R^2\to C$ depending on a parameter $k$, such that if ${\varphi }_1,\dots ,{\varphi }_{n-1}$ are linearly independent solutions of this linear system then $h:=(\ln W({\varphi }_1,\dots ,{\varphi }_{n-1}){)}_x$ 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 $W({\varphi }_1,\dots ,{\varphi }_{n-1})$ 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.

29 pages

References in corpus (2)

• N-dimension Central Affine Curve Flows
• Tau function and Virasoro action for the nxn KdV hierarchy