Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term

The paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient which is usually small in comparison to the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is "supercritical" in the case of two and three space dimensions. Thus the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small inertial coefficient and arbitrary growth rate of the nonlinearity. The present contribution studies the long-time behavior of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers. Namely, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well

Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation

Einstein's relation E=Mc^2 between the energy E and the mass M is the cornerstone of the relativity theory. This relation is often derived in a context of the relativistic theory for closed systems which do not accelerate. By contrast, Newtonian approach to the mass is based on an accelerated motion. We study here a particular neoclassical field model of a particle governed by a nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the nonlinear KG equation and its energy density concentrate at a trajectory, then this trajectory and the energy must satisfy the relativistic version of Newton's law with the mass satisfying Einstein's relation. Therefore the internal energy of a localized wave affects its acceleration in an external field as the inertial mass does in Newtonian mechanics. We demonstrate that the "concentration" assumptions hold for a wide class of rectilinear accelerating motions

Giant Coulomb broadening and Raman lasing on ionic transitions

CW generation of anti-Stokes Raman laser on a number of blue-green argon-ion lines (4p-4s, 4p-3d) has been demonstrated with optical pumping from metastable levels 3d'^2G, 3d^4F. It is found, that the population transfer rate is increased by a factor of 3-5 (and hence, the output power of such Raman laser) owing to Coulomb diffusion in the velocity space. Measured are the excitation and relaxation rates for the metastable level. The Bennett hole on the metastable level has been recorded using the probe field technique. It has been shown that the Coulomb diffusion changes shape of the contour to exponential cusp profile while its width becomes 100 times the Lorentzian one and reaches values close to the Doppler width. Such a giant broadening is also confirmed by the shape of the absorption saturation curve.Comment: RevTex 18 pages, 5 figure

On Asymptotic Completeness of Scattering in the Nonlinear Lamb System, II

We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point using the Inverse Function Theorem in a Banach space. We give the counterexamples showing nonexistence of such trajectories for nonhyperbolic stationary points

Linear superposition in nonlinear wave dynamics

We study nonlinear dispersive wave systems described by hyperbolic PDE's in R^{d} and difference equations on the lattice Z^{d}. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi-Pasta-Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.Comment: New introduction written, style changed, references added and typos correcte

Experimental demonstration of mode structure in ultralong Raman fiber lasers

We present the first experimental demonstration of a resolvable mode structure with spacing c/2nL in the RF spectra of ultralong Raman fiber lasers. The longest ever demonstrated laser cavity (L=84km), RF peaks of ∼100 Hz width and spacing ∼1 kHz have been observed at low intracavity powers. The width of the peaks increases linearly with growing intracavity power and is almost independent of fiber length. © 2007 Optical Society of America