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 $H^1(\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \Delta u = -|u|^4u$. 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, $\dot H^1$-subcritical perturbation $|u|^2u$ 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 $n \geq 3$ 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 $L^2$-critical half-wave equation in one space dimension $$i \partial_t u = D u - |u|^2 u,$$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_* > 0$ such that all $H^{1/2}$ solutions with $\| u \|_{L^2} < M_*$ extend globally in time, while solutions with $\| u \|_{L^2} \geq M_*$ 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 $\| u_0 \|_{L^2} = M_*$. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy $E_0 >0$ and the linear momentum $P_0 \in \R$. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for $L^2$-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page

Uniqueness and Nondegeneracy of Ground States for (−Δ)sQ+Q−Qα+1=0(-\Delta)^s Q + Q - Q^{\alpha+1} = 0 in R\mathbb{R}

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha < \infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ 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 $s=1/2$ and $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$. This result about $L_+$ 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 R1+3R^{1+3}

Consider the focussing cubic nonlinear Schr\"odinger equation in $R^3$: $$i\psi_t+\Delta\psi = -|\psi|^2 \psi.$$ It admits special solutions of the form $e^{it\alpha}\phi$, where $\phi$ is a Schwartz function and a positive ($\phi>0$) solution of $$-\Delta \phi + \alpha\phi = \phi^3.$$ 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 $e^{i(v \cdot + \Gamma)} \phi(\cdot - y, \alpha)$. We prove that any solution starting sufficiently close to a standing wave in the $\Sigma = W^{1, 2}(R^3) \cap |x|^{-1}L^2(R^3)$ 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 $L^2$-subcritical case and adapted by Schlag to the $L^2$-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in $R^3$ for the nonselfadjoint Schr\"odinger operator obtained by linearizing around a standing wave solution.Comment: 56 page