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In this article, we establish scale-invariant Strichartz estimates for the Schr\"odinger equation on arbitrary compact globally symmetric spaces and some bilinear Strichartz estimates on products of rank-one spaces. As applications, we…
We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) flat tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the…
An integrable extension of the well known nonlinear Schroedinger (NLS) equation to a higher space-dimension, recently proposed by us, is investigated, exploring its various important aspects. Focusing on the idea of construction its…
We study the defocusing energy-critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_tu+\Delta u=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u, \qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and…
We present a new proof of global existence and long range scattering, from small initial data, for the one-dimensional cubic gauge invariant nonlinear Schr\"odinger equation, and for Hartree equations in dimension $n \geq 2$. The proof…
We consider the long-time dynamics of focusing energy-critical Schr\"odinger equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$…
We present and analyze a series of conservative diagonally implicit Runge--Kutta schemes for the nonlinear Schr\"odiner equation. With the application of the newly developed invariant energy quadratization approach, these schemes possess…
We prove a general dispersive estimate for a Schroedinger type equation on a product manifold, under the assumption that the equation restricted to each factor satisfies suitable dispersive estimates. Among the applications are the…
We investigate recursive properties of certain p-adic Whittaker functions (of which representation densities of quadratic forms are special values). The proven relations can be used to compute them explicitly in arbitrary dimensions,…
We prove decay with respect to some Lebesgue norms for a class of Schr\"odinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space…
We prove almost global existence for supercritical nonlinear Schr\"odinger equations on the $d$-torus ($d$ arbitrary) on the good geometry selected in part I. This is seen as the Cauchy consequence of I, since the known invariant measure of…
In this paper, we study a tensor-based method for the numerical solution of a class of diffusion--reaction equations defined on spatial domains that admit common curvilinear coordinate representations. Typical examples in 2D include disks…
We discuss bilinear estimates of tempered distributions in the Fourier restriction spaces for the two-dimensional Sch\"odinger equation whose principal part is the d'Alembertian. We prove that the bilinear estimates hold if and only if the…
We show new global well-posedness results for mass-critical nonlinear Schr\"odinger equations on tori in one and two dimensions. For the quintic nonlinear Schr\"odinger equation on the circle we show global well-posedness for initial data…
This note is devoted to establishing two-weight estimates for commutators of singular integrals. We combine multilinearity with product spaces. A new type of two-weight extrapolation result is used to yield the quasi-Banach range of…
We prove global well-posedness for the cubic nonlinear Schr\"odinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold…
We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary rank in the $n$-dimensional Euclidean space as a linear combination of products of Kronecker deltas. By making full use of the symmetries, one…
The magnon Hedin's equations are derived via the Schwinger functional derivative technique, and the resulting self-consistent Green's function method is used to calculate ground state spin patterns and magnetic structure factors for…
Spectral properties of random Schr\"odinger operators are encoded in the average of products of Greens functions. For probability distributions with enough finite moments, the supersymmetric approach offers a useful dual representation.…
We consider the nonlinear Schr\"odinger equation $iu_t + \Delta u= \lambda |u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $\lambda \in {\mathbb C}$ and $\Im \lambda \le 0$. We construct a class of initial values for which the…