Related papers: Dispersive estimates for the Schrodinger equation …

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to…

In the present paper we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators $L=-(\nabla-i\mathbf{a})^TA(\nabla-i\mathbf{a})+V$. The latter class includes, in…

This paper is mainly devoted to study time decay estimates of the higher-order Schr\"{o}dinger type operator $H=(-\Delta)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive…

In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{it\phi(\sqrt{-\Delta})}$, where $\phi: \mathbb{R}^+\to \mathbb{R}$ is smooth away from the origin. Especially, the decay…

Making use of an ${\it ansatz}$ for the eigenfunctions, we obtain an exact closed form solution to the non-relativistic Schr\"{o}dinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$ in two dimensions, where the…

In this paper, we study one-dimensional linear Schr\"odinger equations with multiple moving potentials, known as transfer charge models. Focusing on the non-self-adjoint setting that arises in the study of solitons, we systematically…

The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques…

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS}…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in…

We show that the thermodynamics of the focusing cubic discrete nonlinear Schrodinger equation are exactly solvable in dimensions three and higher. A number of explicit formulas are derived.

In this paper, we establish derivative estimates for the Vlasov-Poisson system with screening interactions around Penrose-stable equilibria on the phase space $\mathbb{R}^d_x\times \mathbb{R}_v^d$, with dimension $d\ge 3$. In particular, we…

We prove optimal high-frequency resolvent estimates for perturbations by large magnetic and electric potentials

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…

This paper is devoted to the well-posedness for dissipative KdV equations $u_t+u_{xxx}+|D_x|^{2\alpha}u+uu_x=0$, $0<\alpha\leq 1$. An optimal bilinear estimate is obtained in Bourgain's type spaces, which provides global well-posedness in…

We present analytically the exact energy bound-states solutions of the Schrodinger equation in $D$-dimensions for a pseudoharmonic potential plus ring-shaped potential of the form $V(r,\theta)=D_{e}(\frac{r}{% r_{e}}-\frac{r_{e}}{r})…

By making use of an ${\it ansatz}$ for the eigenfunction, we obtain the exact solutions to the Schr\"{o}dinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$, both in three dimensions and in two dimensions, where the…

We consider the following nonlinear Schr\"odinger equations with critical growth: \begin{equation} - \Delta u + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial…

This paper proves $L^p$ decay estimates for Schr\"{o}dinger's and wave equations with scalar potentials on three-dimensional Riemannian manifolds. The main result regards small perturbations of a metric with constant negative sectional…

Let $\Delta_\kappa$ be the Dunkl Laplacian on $\mathbb{R}^n$ and $\phi: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. The aim of this manuscript is twofold. First, we study the decay estimate for a class of dispersive semigroup of the…

We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smoothboundary and we describe dispersion forthe semi-classical Schr{\"o}dinger equation with Dirichlet boundary condition. More specifically, we…