430 research outputs found
Analycity and smoothing effect for the coupled system of equations of Korteweg - de Vries type with a single point singularity
We study that a solution of the initial value problem associated for the
coupled system of equations of Korteweg - de Vries type which appears as a
model to describe the strong interaction of weakly nonlinear long waves, has
analyticity in time and smoothing effect up to real analyticity if the initial
data only has a single point singularity at $x=0.
Homoclinic orbits and chaos in a pair of parametrically-driven coupled nonlinear resonators
We study the dynamics of a pair of parametrically-driven coupled nonlinear
mechanical resonators of the kind that is typically encountered in applications
involving microelectromechanical and nanoelectromechanical systems (MEMS &
NEMS). We take advantage of the weak damping that characterizes these systems
to perform a multiple-scales analysis and obtain amplitude equations,
describing the slow dynamics of the system. This picture allows us to expose
the existence of homoclinic orbits in the dynamics of the integrable part of
the slow equations of motion. Using a version of the high-dimensional Melnikov
approach, developed by Kovacic and Wiggins [Physica D, 57, 185 (1992)], we are
able to obtain explicit parameter values for which these orbits persist in the
full system, consisting of both Hamiltonian and non-Hamiltonian perturbations,
to form so-called Shilnikov orbits, indicating a loss of integrability and the
existence of chaos. Our analytical calculations of Shilnikov orbits are
confirmed numerically
Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption
We extend our previous result on the focusing cubic Klein-Gordon equation in
three dimensions to the non-radial case, giving a complete classification of
global dynamics of all solutions with energy at most slightly above that of the
ground state.Comment: 40 page
The dynamics of the 3D radial NLS with the combined terms
In this paper, we show the scattering and blow-up result of the radial
solution with the energy below the threshold for the nonlinear Schr\"{o}dinger
equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u
\tag{CNLS} in the energy space . The threshold is given by the
ground state for the energy-critical NLS: . This
problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The
main difficulty is the lack of the scaling invariance. Illuminated by
\cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the
scaling parameter, then apply it into the scattering theory. Our result shows
that the defocusing, -subcritical perturbation does not
affect the determination of the threshold of the scattering solution of (CNLS)
in the energy space.Comment: 46page
Limiting Carleman weights and anisotropic inverse problems
In this article we consider the anisotropic Calderon problem and related
inverse problems. The approach is based on limiting Carleman weights,
introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean
case. We characterize those Riemannian manifolds which admit limiting Carleman
weights, and give a complex geometrical optics construction for a class of such
manifolds. This is used to prove uniqueness results for anisotropic inverse
problems, via the attenuated geodesic X-ray transform. Earlier results in
dimension were restricted to real-analytic metrics.Comment: 58 page
Global Existence and Uniqueness of Solutions to the Maxwell-Schr{\"o}dinger Equations
The time local and global well-posedness for the Maxwell-Schr{\"o}dinger
equations is considered in Sobolev spaces in three spatial dimensions. The
Strichartz estimates of Koch and Tzvetkov type are used for obtaining the
solutions in the Sobolev spaces of low regularities. One of the main results is
that the solutions exist time globally for large data.Comment: 30 pages. In the revised version, the following modification was
made. (1) A line for dedication was added in the first page. (2) Some lines
were added at the bottom in page 4 and the top in page 5 in the first section
to make the description accurate. (3) Some typographical errors were
corrected throughout the pape
The mixed problem for the Laplacian in Lipschitz domains
We consider the mixed boundary value problem or Zaremba's problem for the
Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on
part of the boundary and Neumann data on the remainder of the boundary. We
assume that the boundary between the sets where we specify Dirichlet and
Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p
and the Dirichlet data is in the Sobolev space of functions having one
derivative in L^p for some p near 1. Under these conditions, there is a unique
solution to the mixed problem with the non-tangential maximal function of the
gradient of the solution in L^p of the boundary. We also obtain results with
data from Hardy spaces when p=1.Comment: Version 5 includes a correction to one step of the main proof. Since
the paper appeared long ago, this submission includes the complete paper,
followed by a short section that gives the correction to one step in the
proo
Nondispersive solutions to the L2-critical half-wave equation
We consider the focusing -critical half-wave equation in one space
dimension where denotes the
first-order fractional derivative. Standard arguments show that there is a
critical threshold such that all solutions with extend globally in time, while solutions with may develop singularities in finite time.
In this paper, we first prove the existence of a family of traveling waves
with subcritical arbitrarily small mass. We then give a second example of
nondispersive dynamics and show the existence of finite-time blowup solutions
with minimal mass . More precisely, we construct a
family of minimal mass blowup solutions that are parametrized by the energy
and the linear momentum . In particular, our main result
(and its proof) can be seen as a model scenario of minimal mass blowup for
-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page
Uniqueness and Nondegeneracy of Ground States for in
We prove uniqueness of ground state solutions for the
nonlinear equation in , where
and for and for . Here denotes the fractional Laplacian
in one dimension. In particular, we generalize (by completely different
techniques) the specific uniqueness result obtained by Amick and Toland for
and in [Acta Math., \textbf{167} (1991), 107--126]. As a
technical key result in this paper, we show that the associated linearized
operator is nondegenerate;
i.\,e., its kernel satisfies .
This result about proves a spectral assumption, which plays a central
role for the stability of solitary waves and blowup analysis for nonlinear
dispersive PDEs with fractional Laplacians, such as the generalized
Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page
A Centre-Stable Manifold for the Focussing Cubic NLS in
Consider the focussing cubic nonlinear Schr\"odinger equation in : It admits special solutions of the form
, where is a Schwartz function and a positive
() solution of The space of
all such solutions, together with those obtained from them by rescaling and
applying phase and Galilean coordinate changes, called standing waves, is the
eight-dimensional manifold that consists of functions of the form . We prove that any solution starting
sufficiently close to a standing wave in the norm and situated on a certain codimension-one local
Lipschitz manifold exists globally in time and converges to a point on the
manifold of standing waves. Furthermore, we show that \mc N is invariant
under the Hamiltonian flow, locally in time, and is a centre-stable manifold in
the sense of Bates, Jones. The proof is based on the modulation method
introduced by Soffer and Weinstein for the -subcritical case and adapted
by Schlag to the -supercritical case. An important part of the proof is
the Keel-Tao endpoint Strichartz estimate in for the nonselfadjoint
Schr\"odinger operator obtained by linearizing around a standing wave solution.Comment: 56 page
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