Hermann Flaschka

Interests

Research

Integrable systems; symplectic and Poisson geometry in convexity theorems, representation theory, and zero-dispersion limits of integrable equations

Teaching

Differential equations, applied analysis, probability theory

Courses

No activities entered.

Scholarly Contributions

Journals/Publications

• Flaschka, H. (2014). Defining a trend for time series using the intrinsic time-scale decomposition. New Journal of Physics, 16, 085004 29pp.
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  Co-authors: Juan Restrepo, Shankar Venkataramani,Darrin Comeau
• Flaschka, H., & Millson, J. (2005). Bending flows for sums of rank one matrices. Canadian Journal of Mathematics, 57(1), 114-158.
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  Abstract: We study certain symplectic quotients of n-fold products of complex projective m-space by the unitary group acting diagonally. After studying nonemptiness and smoothness of these quotients we construct the action-angle variables, defined on an open dense subset, of an integrable Hamiltonian system. The semiclassical quantization of this system reporduces formulas from the representation theory of the unitary group. © Canadian Mathematical Society 2005.
• Bloch, A. M., Flaschka, H., & Ratiu, T. (1993). A Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus. Inventiones Mathematicae, 113(1), 511-529.
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  Abstract: The group of area preserving diffeomorphisms of the annulus acts on its Lie algebra, the globally Hamiltonian vectorfields on the annulus. We consider a certain Hilbert space completion of this group (thinking of it as a group of unitary operators induced by the diffeomorphisms), and prove that the projection of an adjoint orbit onto a "Cartan" subalgebra isomorphic to L2 ([0, 1]) is an infinite-dimensional, weakly compact, convex set, whose extreme points coincide with the orbit, through a certain function, of the "permutation" semigroup of measure preserving transformations of [0, 1]. © 1993 Springer-Verlag.
• Flaschka, H., & Haine, L. (1991). Variétés de drapeaux et réseaux de Toda. Mathematische Zeitschrift, 208(1), 545-556.
• Flaschka, H., Newell, A. C., & Ratiu, T. (1983). Kac-moody lie algebras and soliton equations. II. Lax equations associated with A₁⁽¹⁾. Physica D: Nonlinear Phenomena, 9(3), 300-323.
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  Abstract: The soliton equations associated with sl(2) eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment; they are shown to be generated by a single family of commuting Hamiltonians on a subalgebra of the loop algebra of sl(2). The conserved densities and fluxes of the usual ANKS hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems. The Hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment. © 1983.
• Flaschka, H., Newell, A. C., & Ratiu, T. (1983). Kac-moody lie algebras and soliton equations. III. Stationary equations associated with A₁⁽¹⁾. Physica D: Nonlinear Phenomena, 9(3), 324-332.
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  Abstract: The Hamiltonian structure of stationary soliton equations associated with the AKNS eigenvalue problem is derived in two ways. First, it is shown to arise from the Kostant-Kirillov symplectic structure on a coadjoint orbit in an infinite-dimensional Lie algebra. Second, it is obtained as the restriction to a finite-dimensional manifold of the infinite-dimensional Hamiltonian structure associated with a certain eigenvalue problem polynomial in the eigenvalue parameter. © 1983.
• Ferguson Jr., W. E., Flaschka, H., & McLaughlin, D. W. (1982). Nonlinear normal modes for the Toda Chain. Journal of Computational Physics, 45(2), 157-209.
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  Abstract: The Toda Chain is a nonlinear mass-spring chain which can in principle be integrated analytically by the action-angle theory of classical mechanics. We show that the transformation from physical variables to action variables can be implemented very simply on a computer. We study the correspondence between action variables and the motions of the chain, and find the role of the action variables to be very similar to the role of normal-mode amplitudes in the harmonic chain. This similarity leads to a partly rigorous, partly heuristic normal-mode analysis for the Toda chain. New results are obtained when this normal-mode analysis is applied to certain perturbations of the Toda chain such as the Fermi-Pasta-Ulam chain or the Toda chain with a mass impurity. © 1982.
• Flaschka, H., & Newell, A. C. (1982). The Inverse Monodromy Transform is a Canonical Transformation. North-Holland Mathematics Studies, 61(C), 65-89.
• Holian, B. L., Flaschka, H., & McLaughlin, D. W. (1981). Shock waves in the Toda lattice: Analysis. Physical Review A, 24(5), 2595-2623.
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  Abstract: Known analytical results are used to analyze molecular-dynamics experiments of shock waves in the one-dimensional Toda lattice. (This lattice provides a physically realistic model which contains the hard-sphere and harmonic lattices as limits.) Both explicit solutions and rather general theoretical properties have been employed. The leading edge of the shock front is well represented quantitatively by a single isolated solition. Once compression is properly taken into account, the interior of the shock wave is accurately described by a slowly varying Toda wave train. A sharp transition in the dynamical response exists as the shock strength passes a critical value; this critical value is identified mathematically by the spectral transform for the Toda lattice. Finally, a local spectral transform is used to measure, directly from the numerical data, the wave-train characteristics of the shock profile. © 1981 The American Physical Society.
• Flaschka, H., & Newell, A. C. (1980). Monodromy- and spectrum-preserving deformations I. Communications in Mathematical Physics, 76(1), 65-116.
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  Abstract: A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painlevé equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painlevé equation family is shown to be -d/dx ln(Δ+/Δ-), where Δ+ and Δ- are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system. The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations; additional results will be presented in a sequence of future papers. © 1980 Springer-Verlag.
• Flaschka, H. (1975). On the inverse problem for Hill's operator. Archive for Rational Mechanics and Analysis, 59(4), 293-309.
• Flaschka, H., & Leitman, M. (1975). On semigroups of nonlinear operators and the solution of the functional differential equation x ̇(t) = F(x_t). Journal of Mathematical Analysis and Applications, 49(3), 649-658.
• Flaschka, H. (1974). The Toda lattice. II. Existence of integrals. Physical Review B, 9(4), 1924-1925.
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  Abstract: Following recent computer studies which suggested that the equations of motion of Toda's exponential lattice should be completely Hénon discovered analytical expressions for the constants of the motion. In the present paper, the existence of integrals is proved by a different method. Our approach shows the Toda lattice to be a finite-dimensional analog of the Korteweg-de Vries partial differential equation. Certain integrals of the Toda equations are the counterparts of the conserved quantities of the Korteweg-de Vries equation, and the theory initiated here has been used elsewhere to obtain solutions of the infinite lattice by inverse-scattering methods. © 1974 The American Physical Society.
• Flaschka, H., & Strang, G. (1971). The correctness of the Cauchy problem. Advances in Mathematics, 6(3), 347-379.