We show how to compute form factors, matrix elements of local fields, in the restricted sine-Gordon model, at the reflectionless points, by quantizing solitons. We introduce (quantum) separated variables in which the Hamiltonians are expressed in terms of (quantum) τ-functions. We explicitly describe the soliton wave functions, and we explain how the restriction is related to an unusual hermitian structure. We also present a semi-classical analysis which enlightens the fact that the restricted sine-Gordon model corresponds to an analytical continuation of the sine-Gordon model, intermediate between sine-Gordon and KdV.

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Quantization of solitons and the restricted sine-Gordon model

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra
 
$$\widetilde{\mathfrak{g}\mathfrak{l}}^{ + *} (2,\mathbb{R})$$

 are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e. by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order
 
$$\mathcal{O}(\hbar ^2 )$$

 in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. for each case — in the ambient space ℝ
 n
 , the sphere and the ellipsoid — the Schrodinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lame type.

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Classical and quantum integrable systems in 263-1263-1263-1and separation of variables

Binary constrained flows of soliton equations admitting 2×2 Lax matrices have 2N degrees of freedom, which is twice as many degrees of freedom than in the case of monoconstrained flows. By using the normal method, their Lax matrices directly give rise to first N pairs of canonical separated variables for their separation of variables. We propose a new method to introduce the other N pairs of canonical separated variables and additional separated equations. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies.

Separation of variables for soliton equations via their binary constrained flows

We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectral parameter. We further illustrate our approach by developing the basic algebro-geometric setting for the sGmKdV hierarchy, including Baker-Akhiezer functions, trace formulas, Dubrovin-type equations, and theta function representations for its algebro-geometric solutions. Although we mainly focus on sG-type equations, our formalism also yields the sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type equations combined with the mKdV hierarchy.

A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions

The 2+1-dimensional modified Toda lattice is decomposed into solvable ordinary differential equations with the help of the 1+1-dimensional relativistic Toda lattices. Based on the decomposition and the theory of algebraic curve, the straightening out of various flows, including the continuous flow and discrete flow, is exactly given through the introduced Abel–Jacobi coordinates. The explicit theta function solutions for the 2+1-dimensional modified Toda lattice are obtained explicitly.

Explicit solutions of the 2+1-dimensional modified Toda lattice through straightening out of the relativistic Toda flows

The sine‐Gordon equation is considered in the Hamiltonian framework provided by the Adler–Kostant–Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space g* of a loop algebra g , is parameterized by a finite dimensional symplectic vector space W embedded into g* by a moment map. Real quasiperiodic solutions are computed in terms of theta functions using a Liouville generating function which generates a canonical transformation to linear coordinates on the Jacobi variety of a suitable hyperelliptic curve.

Isospectral flow in loop algebras and quasiperiodic solutions of the sine‐Gordon equation

The sine-Gordon equation is considered in the hamiltonian framework provided by the Adler-Kostant-Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space $\grg^*$ of a loop algebra $\grg$, is parametrized by a finite dimensional symplectic vector space $W$ embedded into $\grg^*$ by a moment map. Real quasiperiodic solutions are computed in terms of theta functions using a Liouville generating function which generates a canonical transformation to linear coordinates on the Jacobi variety of a suitable hyperelliptic curve.

M. A. Wisse

Papers

Isospectral flow in loop algebras and quasiperiodic solutions of the sine‐Gordon equation

Isospectral flow in Loop Algebras and Quasiperiodic Solutions of the Sine-Gordon Equation