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Multi-component Toda lattice hierarchy

T. Takebe, A. Zabrodin

Abstract

We give a detailed account of the N -component Toda lattice hierarchy. This hierarchy is an extended version of the one introduced by Ueno and Takasaki. Our version contains N discrete variables rather than one. We start from the Lax formalism, deduce the bilinear relation for the wave functions from it and then, based on the latter, prove existence of the tau-function. We also show how the multi-component Toda lattice hierarchy is embedded into the universal hierarchy which is basically the multi-component KP hierarchy. At last, we show how the bilinear integral equation for the tau-function can be obtained using the free fermion technique. An example of exact solutions (a multi-component analogue of one-soliton solutions) is given.

Multi-component Toda lattice hierarchy

Abstract

We give a detailed account of the N -component Toda lattice hierarchy. This hierarchy is an extended version of the one introduced by Ueno and Takasaki. Our version contains N discrete variables rather than one. We start from the Lax formalism, deduce the bilinear relation for the wave functions from it and then, based on the latter, prove existence of the tau-function. We also show how the multi-component Toda lattice hierarchy is embedded into the universal hierarchy which is basically the multi-component KP hierarchy. At last, we show how the bilinear integral equation for the tau-function can be obtained using the free fermion technique. An example of exact solutions (a multi-component analogue of one-soliton solutions) is given.
Paper Structure (20 sections, 18 theorems, 382 equations)

This paper contains 20 sections, 18 theorems, 382 equations.

Table of Contents

  1. Introduction
  2. Lax formalism for the multi-component Toda lattice hierarchy
  3. Lax and Zakharov-Shabat equations
  4. Gauge transformations
  5. Linearization: wave operators and wave functions
  6. Wave operators
  7. Wave functions
  8. The bilinear identity
  9. The tau-function
  10. Existence of the tau-function
  11. Tau-function as a universal dependent variable
  12. Bilinear equations of the Hirota-Miwa type
  13. The multi-component Toda lattice from the universal hierarchy
  14. The universal hierarchy
  15. Specification to the multi-component Toda lattice hierarchy
  16. ...and 5 more sections

Key Result

Lemma 2.1

The set of conditions ULQ=UbarLbarQbar:basis ($\alpha=1,\dotsc,N$) is equivalent to for all ${\boldsymbol a}=\{a_1,\dotsc,a_N\}\in{\mathbb Z}^N$.

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.2
  • proof
  • Remark 2.4
  • ...and 30 more