### Abstract

We consider some Lax equations on a periodic lattice with N = 2 sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.

Original language | English (US) |
---|---|

Pages (from-to) | 331-372 |

Number of pages | 42 |

Journal | Transactions of the American Mathematical Society |

Volume | 349 |

Issue number | 1 |

State | Published - 1997 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Transactions of the American Mathematical Society*, vol. 349, no. 1, pp. 331-372.

**On the complete integrability of some lax equations on a periodic lattice.** / Li, Luen-chau.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the complete integrability of some lax equations on a periodic lattice

AU - Li, Luen-chau

PY - 1997

Y1 - 1997

N2 - We consider some Lax equations on a periodic lattice with N = 2 sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.

AB - We consider some Lax equations on a periodic lattice with N = 2 sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.

UR - http://www.scopus.com/inward/record.url?scp=21444438730&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21444438730&partnerID=8YFLogxK

M3 - Article

VL - 349

SP - 331

EP - 372

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -