On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig-Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-Keel-Staffilani-Takaoka-Tao). As an application we prove global well-posedness and scattering in $H^1$ for the energy-critical defocusing initial-value problem (i\partial_t+\Delta_\g)u=u|u|^{4} on the hyperbolic space $H^3$.Comment: 43 pages, references adde

Weighted Strichartz estimates for radial Schr\"odinger equation on noncompact manifolds

We prove global weighted Strichartz estimates for radial solutions of linear Schr\"odinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C^1 potentials decaying like 1/r^2 at infinity

Weierstrass's criterion and compact solitary waves

Weierstrass's theory is a standard qualitative tool for single degree of freedom equations, used in classical mechanics and in many textbooks. In this Brief Report we show how a simple generalization of this tool makes it possible to identify some differential equations for which compact and even semicompact traveling solitary waves exist. In the framework of continuum mechanics, these differential equations correspond to bulk shear waves for a special class of constitutive laws.Comment: 4 page

The tanh and the sine-cosine methods for the complex modified K dV and the generalized K dV equations

AbstractThe complex modified K dV (CMK dV) equation and the generalized K dV equation are investigated by using the tanh method and the sine-cosine method. A variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation. The study reveals the power of the two schemes where each method complements the other

Dispersive estimates with loss of derivatives via the heat semigroup and the wave operator

This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, G\'erard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach, relying only on the heat semigroup in order to understand the analytic connexion between the heat semigroup and the unitary Schr\"odinger group (both related to a same self-adjoint operator). One of the novelty is to forget the endpoint $L^1-L^\infty$ dispersive estimates and to look for a weaker $H^1-BMO$ estimates (Hardy and BMO spaces both adapted to the heat semigroup). This new point of view allows us to give a general framework (infinite metric spaces, Riemannian manifolds with rough metric, manifolds with boundary,...) where Strichartz inequalities with loss of derivatives can be reduced to microlocalized $L^2-L^2$ dispersive properties. We also use the link between the wave propagator and the unitary Schr\"odinger group to prove how short time dispersion for waves implies dispersion for the Schr\"odinger group.Comment: 48 page

Wave and Klein-Gordon equations on hyperbolic spaces

We consider the Klein--Gordon equation associated with the Laplace--Beltrami operator $\Delta$ on real hyperbolic spaces of dimension $n\!\ge\!2$; as $\Delta$ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well--posedness results for the corresponding semilinear equation with low regularity data.Comment: 50 pages, 30 figures. arXiv admin note: text overlap with arXiv:1010.237

Compactons and kink-like solutions of BBM-like equations by means of factorization

In this work, we study the Benjamin-Bona-Mahony like equations with a fully nonlinear dispersive term by means of the factorization technique. In this way we find the travelling wave solutions of this equation in terms of the Weierstrass function and its degenerated trigonometric and hyperbolic forms. Then, we obtain the pattern of periodic, solitary, compacton and kink-like solutions. We give also the Lagrangian and the Hamiltonian, which are linked to the factorization, for the nonlinear second order ordinary differential equations associated to the travelling wave equations.Comment: 10 pages, 8 figure