arXiv subjects

J. Harnad

Publications and source records attributed to J. Harnad.

66 records · Page 4Linked to original sources

arXiv1997

Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations

The Hamiltonian approach to the theory of dual isomonodromic deformations is developed within the framework of rational classical R-matrix structures on loop algebras. Particular solutions to the isomonodromic deformation equations appearing in the computation of correlation functions in integrable quantum field theory models are constructed through the Riemann-Hilbert problem method. The corresponding $τ$-functions are shown to be given by the Fredholm determinant of a special class of integral operators.

solv-int↗

arXiv1997

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed as nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M, whose Poisson quotient, under a parametric family of Hamiltonian group actions, is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem method is used to identify the auxiliary space M with the data defining the integral kernel of the resolvent operator at the endpoints of the curve segments. A second associated isomonodromic family of covariant derivative operators D_z is derived, having rank n=2m, and r finite regular singular points at the values of the exponents defining the kernel of K. This family is similarly embedded into the algebra Lgl_R(n) through a dual parametric family of Poisson quotients of M. The operators D_z are shown to be analogously associated to the integral operator obtained from K through a Fourier-Laplace transform.

solv-int↗

arXiv1996

Bispectral Operators of Rank 1 and Dual Isomonodromic Deformations

A comparison is made between bispectral operator pairs and dual pairs of isomonodromic deformation equations. Through examples, it is shown how operators belonging to rank one bispectral algebras may be viewed equivalently as defining 1-parameter families of rational first order differential operators with matricial coefficients on the Riemann sphere, whose monodromy is trivial. By interchanging the rôles of the two variables entering in the bispectral pair, a second 1-parameter family of operators with trivial monodromy is obtained, which may be viewed as the dual isomonodromic deformation system.

solv-int↗

arXiv1996

Darboux Coordinates on Coadjoint Orbits of Lie Algebras

The method of constructing spectral Darboux coordinates on finite dimensional coadjoint orbits in duals of loop algebras is applied to the one pole case, where the orbit is identified with a coadjoint orbit in the dual of a finite dimensional Lie algebra. The constructions are carried out explicitly when the Lie algebra is $\frak{sl}(2,\bold R),\ \frak{sl}(3, \bold R),$ and $\frak{so}(3, \bold R)$, and for rank two orbits in $\frak{so}(n, \bold R)$. A new feature that appears is the possibility of identifying spectral Darboux coordinates associated to ``dynamical" choices of sections of the associated eigenvector line bundles; i.e. sections that depend on the point within the given orbit.

solv-int↗

arXiv1994

$R$--Matrix Construction of Electromagnetic Models for the Painlevé Transcendents

The Painlevé transcendents $P_{\rom{I}}$--$P_{\rom{V}}$ and their representations as isomonodromic deformation equations are derived as nonautonomous Hamiltonian systems from the classical $R$--matrix Poisson bracket structure on the dual space $\wt{\frak{sl}}_R^*(2)$ of the loop algebra $\wt{\frak{sl}}_R(2)$. The Hamiltonians are obtained by composing elements of the Poisson commuting ring of spectral invariant functions on $\wt{\frak{sl}}_R^*(2)$ with a time--dependent family of Poisson maps whose images are $4$--dimensional rational coadjoint orbits in $\wt{\frak{sl}}_R^*(2)$. Each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time--varying electromagnetic field. The Painlevé equations follow from reduction of these systems by the Hamiltonian flow generated by a second commuting element in the ring of spectral invariants.

hep-th↗

arXiv1994

Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz

The relation between solutions to Helmholtz's equation on the sphere $S^{n-1}$ and the $[{\gr sl}(2)]^n$ Gaudin spin chain is clarified. The joint eigenfuctions of the Laplacian and a complete set of commuting second order operators suggested by the $R$--matrix approach to integrable systems, based on the loop algebra $\wt{sl}(2)_R$, are found in terms of homogeneous polynomials in the ambient space. The relation of this method of determining a basis of harmonic functions on $S^{n-1}$ to the Bethe ansatz approach to integrable systems is explained.

hep-th↗

arXiv1993

Loop Algebra Moment Maps and Hamiltonian Models for the Painleve Transcendants

The isomonodromic deformations underlying the Painlevé transcendants are interpreted as nonautonomous Hamiltonian systems in the dual $\gR^*$ of a loop algebra $\tilde\grg$ in the classical $R$-matrix framework. It is shown how canonical coordinates on symplectic vector spaces of dimensions four or six parametrize certain rational coadjoint orbits in $\gR^*$ via a moment map embedding. The Hamiltonians underlying the Painlevé transcendants are obtained by pulling back elements of the ring of spectral invariants. These are shown to determine simple Hamiltonian systems within the underlying symplectic vector space.

hep-th↗

arXiv1993

Moment Maps to Loop Algebras, Classical R-Matrix and Integrable Systems

A class of Poisson embeddings of reduced, finite dimensional symplectic vector spaces into the dual space $\Lg_R^*$ of a loop algebra, with Lie Poisson structure determined by the classical split $R$--matrix $R=P_+ - P_-$ is introduced. These may be viewed as equivariant moment maps inducing natural Hamiltonian actions of the ``dual'' group $\LG_R = \LGp \times \LGm$ of a loop group $\LG$ on the symplectic space. The $R$--matrix version of the Adler-Kostant-Symes theorem is used to induce commuting flows determined by isospectral equations of Lax type. The compatibility conditions determine finite dimensional classes of solutions to integrable systems of PDE's, which can be integrated using the standard Liouville-Arnold approach. This involves an appropriately chosen ``spectral Darboux'' (canonical) coordinate system in which there is a complete separation of variables. As an example, the method is applied to the determination of finite dimensional quasi-periodic solutions of the sine-Gordon equation.

hep-th↗

arXiv1993

Dual Isomonodromic Deformations and Moment Maps to Loop Algebras

The Hamiltonian structure of the monodromy preserving deformation equations of Jimbo {\it et al } is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to ``dual'' pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve transcendents $P_{V}$ and $P_{VI}$.

hep-th↗

arXiv1992

Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part $\wt{\frak{g}}^+$ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schrödinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation.

hep-th↗

arXiv1992

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.

hep-th↗

arXiv1992

Matrix Nonlinear Schrödinger Equations and Moment Maps into Loop Algebras${}^†$

It is shown how Darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear Schrödinger equations are realized as isospectral Hamiltonian flows. The parametrization follows from a moment map embedding of the symplectic vector space, reduced by suitable group actions, into the dual $\tilde\grg^{+*}$ of the algebra $\tilde\grg^+$ of positive frequency loops in a Lie algebra $\grg$. The resulting phase space is identified with a Poisson subspace of $\tilde\grg^{+*}$ consisting of elements that are rational in the loop parameter. Reduced coordinates associated to the various Hermitian symmetric Lie algebras $(\grg,\grk)$ corresponding to the classical Lie algebras are obtained.

hep-th↗