Kac-moody lie algebras and soliton equations. II. Lax equations associated with A1(1)

H. Flaschka; A. C. Newell; T. Ratiu

doi:10.1016/0167-2789(83)90274-9

Kac-moody lie algebras and soliton equations. II. Lax equations associated with A₁⁽¹⁾

H. Flaschka
, A. C. Newell
, T. Ratiu

Mathematics

Research output: Contribution to journal › Article › peer-review

105 Scopus citations

Abstract

The soliton equations associated with sl(2) eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment; they are shown to be generated by a single family of commuting Hamiltonians on a subalgebra of the loop algebra of sl(2). The conserved densities and fluxes of the usual ANKS hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems. The Hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment.

Original language	English (US)
Pages (from-to)	300-323
Number of pages	24
Journal	Physica D: Nonlinear Phenomena
Volume	9
Issue number	3
DOIs	https://doi.org/10.1016/0167-2789(83)90274-9
State	Published - Dec 1983

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/0167-2789(83)90274-9

Cite this

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title = "Kac-moody lie algebras and soliton equations. II. Lax equations associated with A1(1)",

abstract = "The soliton equations associated with sl(2) eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment; they are shown to be generated by a single family of commuting Hamiltonians on a subalgebra of the loop algebra of sl(2). The conserved densities and fluxes of the usual ANKS hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems. The Hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment.",

author = "H. Flaschka and Newell, \{A. C.\} and T. Ratiu",

note = "Funding Information: There had been a long-standing belief that Lie algebras are important in soliton theory. Such results as could be obtained in the early days generally reflected little more than the fact that, for instance, * Supported in part by DOA Contract DAAG29-82-K-0068 and NSF grant MCS-81-02748A01. **Supported in part by DOA Contract DAAG29-82-K-0068, NSF grant MCS75-07548A01, and ONR Contract N0014-76-C-0867. ***Supported in part by DOA Contract DAAG29-82-K-0068 and NSF grant MCS81-06142. Funding Information: This work is an outgrowth of our efforts to relate the new theory of z-functions and vertex operators to earlier soliton mathematics; the problems might never have been posed had not Flaschka been at the Kyoto University RIMS when the results reported in \textbackslash{}[5\textbackslash{}]w ere being discovered. Flaschka and (through him) the other authors are indebted to E. Date, M. Jimbo, M. Kashiwara, T. Miwa, and M. and Y. Sato for many explanations. We also wish to acknowledge the support of NSF and DOA, which made our collaboration possible.",

year = "1983",

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doi = "10.1016/0167-2789(83)90274-9",

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AU - Flaschka, H.

AU - Newell, A. C.

AU - Ratiu, T.

N1 - Funding Information: There had been a long-standing belief that Lie algebras are important in soliton theory. Such results as could be obtained in the early days generally reflected little more than the fact that, for instance, * Supported in part by DOA Contract DAAG29-82-K-0068 and NSF grant MCS-81-02748A01. **Supported in part by DOA Contract DAAG29-82-K-0068, NSF grant MCS75-07548A01, and ONR Contract N0014-76-C-0867. ***Supported in part by DOA Contract DAAG29-82-K-0068 and NSF grant MCS81-06142. Funding Information: This work is an outgrowth of our efforts to relate the new theory of z-functions and vertex operators to earlier soliton mathematics; the problems might never have been posed had not Flaschka been at the Kyoto University RIMS when the results reported in \[5\]w ere being discovered. Flaschka and (through him) the other authors are indebted to E. Date, M. Jimbo, M. Kashiwara, T. Miwa, and M. and Y. Sato for many explanations. We also wish to acknowledge the support of NSF and DOA, which made our collaboration possible.

PY - 1983/12

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N2 - The soliton equations associated with sl(2) eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment; they are shown to be generated by a single family of commuting Hamiltonians on a subalgebra of the loop algebra of sl(2). The conserved densities and fluxes of the usual ANKS hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems. The Hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment.

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Kac-moody lie algebras and soliton equations. II. Lax equations associated with A₁⁽¹⁾

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