dorsal/arxiv
View SchemaGeneralized Jacobians of spectral curves and completely integrable systems
| Authors | Lubomir Gavrilov |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9810001 |
| URL | https://arxiv.org/abs/solv-int/9810001 |
| Journal | Math. Zeitschrift, 230, 487-508 (1999) |
Abstract
Consider an ordinary differential equation which has a Lax pair representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold {A(x): det(A(x)-y I)= P(x,y)} of this Lax pair is an affine part of a non-compact commutative algebraic group---the generalized Jacobian of the spectral curve {(x,y): P(x,y)=0} with its points at "infinity" identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pairon the generalized Jacobian is translation--invariant. We provide two examples in which the above result applies.
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"abstract": "Consider an ordinary differential equation which has a Lax pair\nrepresentation A\u0027(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a\nfixed regular leading coefficient and the matrix B(x) depends only onA(x). Such\nan equation can be considered as a completely integrable complex Hamiltonian\nsystem. We show that the generic complex invariant manifold {A(x): det(A(x)-y\nI)= P(x,y)} of this Lax pair is an affine part of a non-compact commutative\nalgebraic group---the generalized Jacobian of the spectral curve {(x,y):\nP(x,y)=0} with its points at \"infinity\" identified. Moreover, for suitable\nB(x), the Hamiltonian vector field defined by the Lax pairon the generalized\nJacobian is translation--invariant. We provide two examples in which the above\nresult applies.",
"arxiv_id": "solv-int/9810001",
"authors": [
"Lubomir Gavrilov"
],
"categories": [
"solv-int",
"nlin.SI"
],
"journal_ref": "Math. Zeitschrift, 230, 487-508 (1999)",
"title": "Generalized Jacobians of spectral curves and completely integrable systems",
"url": "https://arxiv.org/abs/solv-int/9810001"
},
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