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Commuting difference operators, spinor bundles and the asymptotics of pseudoorthogonal polynomials with respect to varying complex weights
, 2006
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Duality for the general isomonodromy problem
 J. Geom. Phys
"... By an extension of Harnad’s and Dubrovin’s ‘duality ’ constructions, the general isomonodromy problem studied by Jimbo, Miwa, and Ueno is equivalent to one in which the linear system of differential equations has a regular singularity at the origin and an irregular singularity at infinity (both reso ..."
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By an extension of Harnad’s and Dubrovin’s ‘duality ’ constructions, the general isomonodromy problem studied by Jimbo, Miwa, and Ueno is equivalent to one in which the linear system of differential equations has a regular singularity at the origin and an irregular singularity at infinity (both resonant). The paper looks at this dual formulation of the problem from two points of view: the symplectic geometry of spaces associated with the loop group of the general linear group, and a generalization of the selfdual YangMills equations.
CRM3239 Effective inverse spectral problem for rational Lax
, 705
"... We reconstruct a rational Lax matrix of size R + 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy–like kernel (a bidifferential of biweight (1−ν,ν)) we provide a residueformula for the entries of the Lax matrix in ..."
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We reconstruct a rational Lax matrix of size R + 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy–like kernel (a bidifferential of biweight (1−ν,ν)) we provide a residueformula for the entries of the Lax matrix in terms of bases of dual differentials of weights ν,1−ν respectively. All objects are described in the most explicit terms using Theta functions. Via a sequence of “elementary twists”, we construct sequences of Lax matrices sharing the same spectral curve and polar structure and related by conjugations by rational matrices. Particular choices of elementary twists lead to construction of sequences of Lax matrices related to finite–band recurrence relations (i.e. difference operators) sharing the same shape. Recurrences of this kind are satisfied by several types of orthogonal and biorthogonal polynomials. The relevance
Centre de recherches mathématiques
, 705
"... The nonlinear steepest descent method for ranktwo systems relies on the notion of gfunction. The applicability of the method ranges from orthogonal polynomials (and generalizations) to Painlevé transcendents, and integrable wave equations (KdV, NonLinear Schrödinger,etc.). For the case of asymptot ..."
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The nonlinear steepest descent method for ranktwo systems relies on the notion of gfunction. The applicability of the method ranges from orthogonal polynomials (and generalizations) to Painlevé transcendents, and integrable wave equations (KdV, NonLinear Schrödinger,etc.). For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast the requirements for the Cauchytransform of the equilibrium measure into a problem of algebraic geometry and harmonic analysis and completely solve the existence and uniqueness issue without relying on the minimization of a functional. This addresses and solves also the issue of the “free boundary problem”, determining implicitly the curves where the zeroes of the orthogonal polynomials accumulate in the limit of large degrees and the support of the measure. The relevance to the quasi—linear Stokes phenomenon for Painlevé equations is indicated. A numerical algorithm to find these curves in some cases is also explained. Technical note: the animations included in the file can be viewed using Acrobat Reader 7 or higher. Mac users should also install a QuickTime plugin called Flip4Mac. Linux users can extract the embedded animations and play them with an external program like VLC or MPlayer. All trademarks are owned by the respective companies.
CRM3239 Effective inverse spectral problem for rational Lax
, 705
"... We reconstruct a rational Lax matrix of size R + 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy–like kernel (a bidifferential of biweight (1−ν,ν)) we provide a residueformula for the entries of the Lax matrix in ..."
Abstract
 Add to MetaCart
(Show Context)
We reconstruct a rational Lax matrix of size R + 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy–like kernel (a bidifferential of biweight (1−ν,ν)) we provide a residueformula for the entries of the Lax matrix in terms of bases of dual differentials of weights ν,1−ν respectively. All objects are described in the most explicit terms using Theta functions. Via a sequence of “elementary twists”, we construct sequences of Lax matrices sharing the same spectral curve and polar structure and related by conjugations by rational matrices. Particular choices of elementary twists lead to construction of sequences of Lax matrices related to finite–band recurrence relations (i.e. difference operators) sharing the same shape. Recurrences of this kind are satisfied by several types of orthogonal and biorthogonal polynomials. The relevance