On the global wellposedness of the Klein-Gordon equation for initial data in modulation spaces

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u|^{\alpha−1}u$, where $\alpha\in\left[1,\frac{d}{d−2}\right]$, in dimension $d\ge3$ with initial data in $M^1_{p,p\u27}(\mathbb{R}^d)\times M_{p,p\u27}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is an application of the high-low method described by Bourgain in [1] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces

Higher order NLS with anisotropic dispersion and modulation spaces: a global existence and scattering result

In this paper we transfer a small data global existence and scattering result by Wang and Hudzik to the more general case of modulation spaces $M_{p, q}^s(\mathbb{R}^d)$ where $q = 1$ and $s \geq 0$ or $q \in (1, \infty]$ and $s > \frac{d}{q'}$ and to the nonlinear Schr\"odinger equation with higher order anisotropic dispersion.Comment: 9 page

Interference and Throughput in Aloha-based Ad Hoc Networks with Isotropic Node Distribution

We study the interference and outage statistics in a slotted Aloha ad hoc network, where the spatial distribution of nodes is non-stationary and isotropic. In such a network, outage probability and local throughput depend on both the particular location in the network and the shape of the spatial distribution. We derive in closed-form certain distributional properties of the interference that are important for analyzing wireless networks as a function of the location and the spatial shape. Our results focus on path loss exponents 2 and 4, the former case not being analyzable before due to the stationarity assumption of the spatial node distribution. We propose two metrics for measuring local throughput in non-stationary networks and discuss how our findings can be applied to both analysis and optimization.Comment: 5 pages, 3 figures. To appear in International Symposium on Information Theory (ISIT) 201

The global Cauchy problem for the NLS with higher order anisotropic dispersion

We use a method developed by Strauss to obtain global wellposedness results in the mild sense for the small data Cauchy problem in modulation spaces $M^s_{p,q}(\mathbb{R}^d)$, where $q = 1$ and $s\ge 0$ or $q \in (1, \infty]$ and $s > \frac{d}{q\u27}$ for a nonlinear Schrödinger equation with higher order anisotropic dispersion and algebraic nonlinearities

On the global well-posedness of the quadratic NLS on L2(R)+H1(R)L^2(\mathbb{R})+H^1(\mathbb{R})

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $|u|^{\alpha-1}$ for $\alpha \in [2, 5]$ and initial data $u_0 ∈ L^2(\mathbb{R})+H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity $(\alpha = 2)$ we obtain unconditional global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb{R})+H^1(\mathbb{T}))$ via Gronwall’s inequality

Localwell-posedness for the nonlinear Schrödinger equation in modulation spaces Ms p;q(Rd)

We show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation on modulation spaces Msp;q(Rd) for d 2 N, 1 p; q 1 and s > d 1 1 q for q > 1 or s 0 for q = 1. This improves [4, Theorem 1.1] by Bényi and Okoudjou where only the case q = 1 is considered. Our result is based on the algebra property of modulation spaces with indices as above for which we give an elementary proof via a new Hölder-like inequality for modulation spaces

Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces Mp,qs(Rd)∩M∞,1(Rd)M^s_{p,q}(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, implicitly contained in [STW11], of the intersection $M^s_{p,q}(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)$ for $d\in\mathbb{N}$, $p, q\in [1,\infty]$ and $s\ge0$. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves [BO09, Theorem 1.1] by Bényi and Okoudjou, where only the case $q=1$ is considered, and closes a gap in the literature. If $q>1$ and $s>d(1-\frac{1}{q})$ or if $q=1$ and $s\geq0$ then $M^s_{p,q}(\mathbb{R}^d) \hookrightarrow M_{\infty,1}(\mathbb{R}^d)$ and the above intersection is superfluous. For this case we also obtain a new Hölder-type inequality for modulation spaces