A “resonance” here is defined to take place iff W (0) = 0 where W (λ) is the Wronskian of the two Jost solutions at energy λ2, see the following section. It is known that the spectrum of H is purely absolutely continuous on (0,∞) under our assumptions (V ∈ L1(R) suffices for that) so that Pac is the same as the projection onto the orthogonal complement of the bound states. For the case of three...

A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations....

For a large class of complete, non-compact Riemannian manifolds, (M, g), with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator:

The Korteweg de Vries (KdV) equation with small dispersion is a model for the formation and propagation of dispersive shock waves in one dimension. Dispersive shock waves in KdV are characterized by the appearance of zones of rapid modulated oscillations in the solution of the Cauchy problem with smooth initial data. The modulation in time and space of the amplitudes, the frequencies and the wa...

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in periodic setting. Our main hypotheses are both that operator behaves high frequencies as Fourier multiplier by i|ξ|αξ, 1≤α≤2, and nonlinear term is of form ∂xf(u) where f sum an entire series infinite radius convergence. Under these conditions, we prove unconditional local well-posedness Hs(T)...

The KdV equation with small dispersion is a model for the formation and propagation of dispersive shock waves. Dispersive shock waves are characterized by the appearance of modulated oscillations nearby the breaking point. The modulation in time and space of the amplitude, the frequencies and the wave-numbers of these oscillations is described by the g-phase Whitham equations. We study the init...