Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner transform (WT) is a quadratic transform that takes an oscillatory function u (x): ℝⁿ ↦ ℂ^d to a phase-space density W (x, k) = W [u ](x, k): ℝ²ⁿ ↦ ℂ^{d ×d} , resolving it over an additional set of 'wavenumber' variables. The WT and its variations have been heavily used in quantum mechanics, semiconductors, homogenization of wave equations, timefrequency analysis, signal processing, pseudodifferential operators etc. The WT however has a fundamental difficulty: WTs exhibit artifacts, collectively known as ‘interference terms’, and can be arbitrarily more complicated than the original wavefunction. A very successful, well established way to go around this is using the Wigner measures (WMs), a semiclassical approximation to the WT. We propose a different approach, namely smoothing the WT with an appropriate kernel. Such smoothed WTs (SWTs) have been used with great success in signal processing. They have not been used in the treatment of PDEs, a fundamental obstacle being the lack of exact equations governing their evolution. We present the machinery which allows the coarse-scale reformulation of a broad class of wave problems in terms of the SWT, along with numerical experiments which clearly show the validity and applicability of the method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner transform (WT) has been extensively used in the formulation of phase-space models for a variety of wave propagation problems including high-frequency limits, nonlinear and random waves. It is well known that the WT features counterintuitive ‘interference terms,’ which often make computation impractical. In this connection, we propose the use of the smoothed Wigner transform (SWT), and derive new, exact equations for it, covering a broad class of wave propagation problems. Equations for spectrograms are included as a special case. The ‘taming’ of the interference terms by the SWT is illustrated, and an asymptotic model for the Schrödinger equation is constructed and numerically verified.     

On the completeness of eigenfunctions of some higher order operators

The completeness of diverse eigenfunction systems is of interest, e.g., in connection with their use for expansion purposes. Whereas second-order operators have been treated extensively, it is more difficult to find valid proofs even for comparatively simple fourth- and higher-order operators. A criterion is developed that allows known properties of second-order differential operators to be used as a basis for conclusions regarding the completeness of the eigenfunctions of related fourth-order operators. The completeness of the “flatclamped-plate modes” is proved as an example, and it is demonstrated that the detailed form of the “conditions of finiteness at the singular point” can be crucial for the definition of operators corresponding to differential expressions with singular points.     

Composition and spectral invariance of pseudodifferential Operators on Modulation Spaces

We introduce new classes of Banach algebras of pseudodifferential operators with symbols in certain modulation spaces and investigate their composition and the functional calculus. Operators in these algebras possess the spectral invariance property on the associated family of modulation spaces. These results extend and contain Sjöstrand's theory, and they are obtained with new phase-space methods instead of “hard analysis”.     

Remarques sur les mesures de Wigner

We compute the Wigner measures associated to solutions of semi-classical nonlinear Schrödinger equations. These solutions focus at a point. Outside the caustic, the measures “smooth” the nonlinearity. In the critical cases, a scattering operator describes the jump of Wigner measures at the focus. We show that the problem is ill-posed in terms of Wigner measures.     

Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states

Contrary to the finite dimensional case, Weyl and Wick quantizations are no more asymptotically equivalent in the infinite dimensional bosonic second quantization. Moreover neither the Weyl calculus defined for cylindrical symbols nor the Wick calculus defined for polynomials are preserved by the action of a nonlinear flow. Nevertheless taking advantage carefully of the information brought by these two calculuses in the mean field asymptotics, the propagation of Wigner measures for general states can be proved, extending to the infinite dimensional case a standard result of semiclassical analysis.     

The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings

Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., they have or “almost have” fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover, we present various examples illustrating that our bounds are sharp.     

Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

Semiclassical second microlocal propagation of regularity and integrable systems

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold X with respect to a Lagrangian submanifold of T ^* X. The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of T ^*? ⁿ , and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to Hörmander’s theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g., eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.     

On the solution of algebraic equations by the decomposition method

The decomposition method (G. Adomian, “Stochastic Systems,” Academic Press, New York, 1983) developed to solve nonlinear stochastic differential equations has recently been generalized to nonlinear (and/or) stochastic partial differential equations, systems of equations, and delay equations and applied to diverse applications. As pointed out previously (see reference above) the methodology is an operator method which can be used for nondifferential operators as well. Extension has also been made to algebraic equations involving real or complex coefficients. This paper deals specifically with quadratic, cubic, and general higher-order polynomial equations and negative, or nonintegral powers, and random algebraic equations. Further work on this general subject appears elsewhere (G. Adomian, “Stochastic Systems II,” Academic Press, New York, in press).     

High-frequency propagation for the Schrödinger equation on the torus

The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schrödinger equation on the standard d-dimensional torus Td. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T^*Td. This construction is related to that of the so-called two-microlocal semiclassical measures. We prove that any limit μ of the Wigner distributions corresponding to solutions to the Schrödinger equation on the torus is completely determined by the limits of both the Wigner distribution and the resonant Wigner distribution of the initial data; moreover, μ follows a propagation law described by a family of density-matrix Schrödinger equations on the periodic geodesics of Td. Finally, we present some connections with the study of the dispersive behavior of the Schrödinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schrödinger equation on T² are absolutely continuous with respect to the Lebesgue measure.     

An analysis of the Wigner–Poisson problem with inflow boundary conditions

We present a study of the Wigner–Poisson problem in a bounded spatial domain with non-homogeneous and time-dependent “inflow” boundary conditions. This system of nonlinearly coupled equations is a mathematical model for quantum transport of charges in a semiconductor with external contacts. We prove well-posedness of the linearized n-dimensional problem as well as existence and uniqueness of a global-in-time, regular solution of the one-dimensional nonlinear problem.     

IDENTIFICATION OF EXTRAPOLATION SPACES FOR UNBOUNDED OPERATORS

Abstract

Abstract extrapolation spaces for strongly continuous semigroups of linear operators on Banach spaces have been constructed by various methods (see, e.g., [Am (1988)], [DaP-Gr (1984)], [Na (1983)], [Ne (1992)], [Wa (1986)]). Usually they appear as “artefacts” used in some intermediate step in order to solve the Cauchy problem on the original space. Only in a few cases (see the papers by the Dutch school on X ^⊙*, e.g., [Ne (1992)]), and in sharp contrast to the situation for interpolation spaces (see, e.g., [Gr (1969)], [DiB (1991)], [Lu (1985)], [Ac-Te (1987)]), the extrapolation spaces have been identified in a concrete way. It is our intention to fill this gap and subsequently to give an application of the extrapolation method to a perturbation problem.     

Semiclassical Limit of Nonlinear Schrodinger Equation (II)

In this paper, we use the Wigner measure approach to study the semiclassical limit of nonlinear Schrödinger equation in small time. We prove that: the limits of the quantum density: ρ^∈ =: |ψ^∈|² and the quantum momentum: J^∈ =: ∈Im(\overline{ψ^∈}∇ψ^∈) satisfy the compressible Euler equations before the formation of singularities in the limit system.     

Semiclassical dynamics and coherent soliton condensates in self-focusing nonlinear media with periodic initial conditions

The semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of ICs, referred to as “periodic single-lobe” potentials, share the same qualitative features, which also coincide with those of solutions arising from localized ICs. The spectrum of the associated scattering problem in each of these cases is then numerically computed, and it is shown that such spectrum is confined to the real and imaginary axes of the spectral variable in the semiclassical limit. This implies that all nonlinear excitations emerging from the input have zero velocity, and form a coherent nonlinear condensate. Finally, by employing a formal Wentzel-Kramers-Brillouin expansion for the scattering eigenfunctions, asymptotic expressions for the number and location of the bands and gaps in the spectrum are obtained, as well as corresponding expressions for the relative band widths and the number of “effective solitons.” These results are shown to be in excellent agreement with those from direct numerical computation of the eigenfunctions. In particular, a scaling law is obtained showing that the number of effective solitons is inversely proportional to the small dispersion parameter.     

The nisio semigroup for controlled diffusions with partial observations

A one parameter semigroup of nonlinear operators on an appropriate Banach space is constructed in the spirit of Nisio for controlled diffusions with partial observations. The method is based upon considering an equivalent problem of controlling a measure-valued process representing the conditional law of the state given past observations. The latter evolves according to the usual equations of nonlinear filtering. By considering an appropriate augmentation of the class of controls, it is shown that the “minimum cost” operators associated with this control problem indeed form a semigroup of nonlinear contractions on the space of bounded continuous real-valued functions on the state space of the above measure-valued process.     

Periodic waves in rotating plane Couette flow

A class of nonlinear equations of Navier-Stokes type of the form (**) $$\frac{{dw}}{{dt}} + L_0 w + (\lambda - \lambda _0 )L_1 w + \gamma \lambda (M_1 w + M_2 w) + \gamma ^2 L_3 (\lambda ,\gamma )w + B(w) = 0$$ is investigated, where λ is a “load” parameter (i.e., a Reynolds number), γ is a “structure” parameter,L _o L ₁,M ₁,M ₂ andL ₃(λ, γ) are linear operators, andB is a quadratic operator. An equation of the form (**) describes a variety of spiral flow problems including rotating plane Couette flow which is studied here in detail. Under suitable hypotheses on the operators in (**), it is shown that Hopf bifurcation occurs for γ sufficiently small. In the problem of rotating plane Couette flow, by determining the sign of the real part of a certain “cubic” coefficient, it is shown, in addition, that the bifurcating periodic orbits are supercritical and asymptotically stable, and correspond to periodic waves.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: II. The Covariant Mean-Field Approximation

A covariant kinetic equation for the matrix Wigner function is derived in the mean-field approximation from a general kinetic equation for the fermionic subsystem of a quantum electrodynamic plasma. We show that in the semiclassical limit, the equations for the components of the Wigner function can be transformed into closed kinetic equations for the Lorentz-invariant distribution functions of particles and antiparticles.