Dual Isomonodromic Problems and Whitham Equations
نویسنده
چکیده
The author’s recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter ǫ. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the SeibergWitten solutions of low energy supersymmetric gauge theories.
منابع مشابه
Spectral Curves and Whitham Equations in Isomonodromic Problems of Schlesinger Type
The Schlesinger equation is reformulated to include a small parameter ǫ. In the small-ǫ limit, solutions of this isomonodromic problem are expected to behave like a slowly modulated finite-gap solution of an isospectral problem. The modulation is caused by slow deformations of the spectral curve of the finite-gap solution. A modulation equation of this slow dynamics is derived by a heuristic me...
متن کاملThe symplectic and twistor geometry of the general isomonodromic deformation problem
Hitchin’s twistor treatment of Schlesinger’s equations is extended to the general isomonodromic deformation problem. It is shown that a generic linear system of ordinary differential equations with gauge group SL(n,C) on a Riemann surface X can be obtained by embedding X in a twistor space Z on which sl(n,C) acts. When a certain obstruction vanishes, the isomonodromic deformations are given by ...
متن کاملThe geometry of dual isomonodromic deformations
The JMMS equations are studied using the geometry of the spectral curve of a pair of dual systems. It is shown that the equations can be represented as time-independent Hamiltonian flows on a Jacobian bundle.
متن کاملThe twistor theory of Whitham hierarchy
We have generalized the approach in of Dunajski, Mason and Tod [10] and established a 1-1 correspondence between a solution of the universal Whitham hierarchy [23] and a twistor space. The twistor space consists of a complex surface and a family of complex curves together with a meromorphic 2-form. The solution of the Whitham hierarchy is given by deforming the curve in the surface. By treating...
متن کاملSpectral curves and discrete Painlevé equations
It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of differential and difference systems. The characteristic equations in the case of the associated linear problems for various discrete Painlevé equations is biquadratic ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1998