Symplectic forms in the theory of Solitons

Abstract

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω = 1 2 Res ∞ < Ψ * 0 δL ∧ δΨ 0 > dk. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.* Research supported in part by the National Science Foundation under grant DMS-95-05399. g i=1 dλ(z i ) when finite-gap solitons are imbedded in the space of doubly periodic operators. Here we show that it is a symplectic form in its own right on L(b), and that with respect to this form, the hierarchy of 2D flows is Hamiltonian. Their Hamiltonians are shown to be nH n+m , where H s are the coefficients of the expansion of the quasi-momentum in terms of the quasi-energy.• Our formalism is powerful enough to encompass many diverse symplectic structures for 1D solitons. For example, ω reduces to the Gardner-Faddeev-Zakharov symplectic

Vector Bundles and Lax Equations on Algebraic Curves

“…Moreover, a'priori it is not clear, why all the systems constructed above are Hamiltonian. In this section we show that the general algebraic approach to the Hamiltonian theory of the Lax equations proposed in [9,10] and developed in [11] is evenly applicable to the Lax equations on the Riemann surfaces.…”

“…Moreover, a'priori it's not clear, why the Lax equations are Hamiltonian. In Section 4 we clarify this problem using the approach to the Hamiltonian theory of soliton equations proposed in [9,10,11]. It turns out that for D = K the universal two-form which is expressed in terms of the Lax matrix and its eigenvectors coincides with canonical symplectic structure on the cotangent bundle T * (M).…”

Integrable chains on algebraic curves

“…In order to analyze this problem we first find the Darboux variables for ω. From (4.8) it follows that ω can be represented in the form: 9) where ϕ k are the coordinates on J(Γ), corresponding to a choice of a-and b-cycles onΓ with the canonical matrix of intersections, and …”

“…In this section we apply the general algebraic approach to the Hamiltonian theory of the Lax equations proposed in [8,9], and developed in [13], to the Lax equations for periodic chains on the algebraic curves. As it was mentioned in the introduction, this approach is based on the existence of two universal two-forms on a space of meromorphic matrix-function.…”

“…The alternative approach to the Hamiltonian theory of the soliton equations was developed in [8,9]. It is based on the existence of some universal two-form defined on a space of meromorphic matrix-functions.…”