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Negative flows and non-autonomous reductions of the Volterra lattice

V. E. Adler

Abstract

We study reductions of the Volterra lattice corresponding to stationary equations for the additional, noncommutative subalgebra of symmetries. It is shown that, in the case of general position, such a reduction is equivalent to the stationary equation for a sum of the scaling symmetry and the negative flows, and is written as $(m+1)$-component difference equations of the Painlevé type generalizing the dP$_1$ and dP$_{34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the Bäcklund transformations which form the $\mathbb{Z}^m$ lattice.

Negative flows and non-autonomous reductions of the Volterra lattice

Abstract

We study reductions of the Volterra lattice corresponding to stationary equations for the additional, noncommutative subalgebra of symmetries. It is shown that, in the case of general position, such a reduction is equivalent to the stationary equation for a sum of the scaling symmetry and the negative flows, and is written as -component difference equations of the Painlevé type generalizing the dP and dP equations. For these reductions, we present the isomonodromic Lax pairs and derive the Bäcklund transformations which form the lattice.
Paper Structure (5 sections, 7 theorems, 77 equations)

This paper contains 5 sections, 7 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Negative flows of the Volterra lattice
  3. Construction of the non-autonomous reductions
  4. Darboux--Bäcklund transformations
  5. Conclusion

Key Result

Proposition 1

Equation (uy) is consistent with derivations $D_x$ and $D_t$ defined by (ux), (yx) and (ut), (yt).

Theorems & Definitions (12)

  • Definition
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • ...and 2 more