Properties of periodic solutions near their oscillation threshold for a class of hyperbolic partial differential equations with localized nonlinearity

The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure interaction. It also encompasses particular types of delay differential equations. These systems undergo a bifurcation with the appearance of a small amplitude periodic regime. Assuming a certain regularity of the oscillating solution, several of its properties around the bifurcation are given: bifurcation point, dependence of both the amplitude and period with respect to the bifurcation parameter, and law of decrease of the Fourier series components. All the properties of the standard Hopf bifurcation in the non-hyperbolic case are retrieved. In addition, this study is based on a Fourier domain analysis and the harmonic balance method has been extended to the class of infinite dimensional problems hereby considered. Estimates on the errors made if the Fourier series is truncated are provided.Comment: 20 page

A stability criterion for high-frequency oscillations

We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initial-value problems issued from highly-oscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that hyperbolicity is preserved around resonances. If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time. The amplification mechanism is based on the observation that in frequency space, resonances correspond to points of weak hyperbolicity. At such points, the behavior of the system depends on the lower order terms through the compatibility condition. The analysis relies, in the unstable case, on a short-time Duhamel representation formula for solutions of zeroth-order pseudo-differential equations. Our examples include coupled Klein-Gordon systems, and systems describing Raman and Brillouin instabilities.Comment: Final version, to appear in M\'em. Soc. Math. F

Recurrent Acceleration in Dilaton-Axion Cosmology

A class of Einstein-dilaton-axion models is found for which almost all flat expanding homogeneous and isotropic universes undergo recurrent periods of acceleration. We also extend recent results on eternally accelerating open universes.Comment: 8 pages, 7 figures. minor changes. Version 4 corrects a figure captio

A new class of Fermionic Projectors: M{\o}ller operators and mass oscillation properties

Recently, a new functional analytic construction of quasi-free states for a self-dual CAR algebra has been presented in \cite{Felix2}. This method relies on the so-called strong mass oscillation property. We provide an example where this requirement is not satisfied, due to the nonvanishing trace of the solutions of the Dirac equation on the horizon of Rindler space, and we propose a modification of the construction in order to weaken this condition. Finally, a connection between the two approaches is built.Comment: 21 pages, accepted for publication in Letters in Mathematical Physics ( 998

Oscillation and asymptotic behavior for a class of delay parabolic differential

AbstractSome comparative theorems are given for the oscillation and asymptotic behavior for a class of high order delay parabolic differential equations of the form ∂n(u(x,t)−p(t)u(x,t−τ))∂tn−a(t)△u+c(x,t,u)+∫abq(x,t,ξ)f(u(x,g1(t,ξ)),…,u(x,gl(t,ξ)))dσ(ξ)=0,(x,t)∈Ω×R+≡G, where n is an odd integer, Ω is a bounded domain in Rm with a smooth boundary ∂Ω, and △ is the Laplacian operation with three boundary value conditions. Our results extend some of those of [P. Wang, Oscillatory criteria of nonlinear hyperbolic equations with continuous deviating arguments, Appl. Math. Comput. 106 (1999), 163–169] substantially

A note on string solutions in AdS_3

We systematically search for classical open string solutions in AdS_3 within the general class expressed by elliptic functions (i.e., the genus-one finite-gap solutions). By explicitly solving the reality and Virasoro conditions, we give a classification of the allowed solutions. When the elliptic modulus degenerates, we find a class of solutions with six null boundaries, among which two pairs are collinear. By adding the S^1 sector, we also find four-cusp solutions with null boundaries expressed by the elliptic functions.Comment: 17 pages, 1 figure; (v2) added 1 figure and discussion on solutions with 6 null boundaries; (v3) corrected equation numbers; (v4) added comment