Related papers: Schr\"odinger-type equations in Gelfand-Shilov spa…
We consider the Cauchy problem for Schr\"odinger type operators. Under a suitable decay assumption on the imaginary part of the first order coefficients we prove well-posedness of the Cauchy problem in Gelfand-Shilov classes. We also…
Using the theory of almost conserved energies and the ``I-method'' developed by Colliander, Keel, Staffilani, Takaoka and Tao, we prove that the initial value problem for a higher order Schr\"odinger equation is globally well-posed in…
We consider the Schr\"odinger map initial value problem into the sphere in 2+1 dimensions with smooth, decaying, subthreshold initial data. Assuming an a priori $L^4$ boundedness condition on the solution, we prove that the Schr\"odinger…
Applying an Abstract Interpolation Lemma, we can show persistence of solutions of the initial value problem to higher order nonlinear Schr\"odinger equation, also called Airy-Schr\"odinger equation, in weighted Sobolev spaces…
We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schroedinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes…
In this paper we discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schr\"odinger on the sphere $S^2$. Exploring suitable a priori estimates, we prove the existence of solution for…
The initial value problem is considered for a higher order nonlinear Schr\"odinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional…
We study the wave front set of the solutions of the initial value problem for nonlinear Schr\"{o}dinger equations via wave packet transform. We give an sufficient condition which assures that the solutions is in Sobolev space of order s in…
We consider the Cauchy problem for a $3$-evolution operator $P$ with $(t,x)$-depending coefficients and complex valued lower order terms. We assume the initial data to be Gevrey regular and to admit an exponential decay at infinity, that…
In this paper, we establish refined Strichartz estimates for higher-order Schr\"odinger equations with initial data exhibiting partial regularity. By partial regularity, we mean that the initial data are not required to have full Sobolev…
We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail…
We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…
We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained.
We consider the defocusing fourth-order nonlinear Schr\"{o}dinger equation with potential \[ i\partial_t u + \Delta^2 u + Vu + \lambda |u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), \] in dimensions $n \ge 5$. In the…
Let X be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schr\"odinger equation on X with square integrable initial condition f is identically zero at all times t whenever f and the…
We study local and global well-posedness of the initial value problem for the Schr\"odinger-Debye equation in the \emph{periodic case}. More precisely, we prove local well-posedness for the periodic Schr\"odinger-Debye equation with…
In this article we will study the initial value problem for some Schr\"odinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in…
The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^{s}({\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D…
We consider the Schr\"{o}dinger map initial-value problem in dimension two or greater. We prove that the Schr\"{o}dinger map initial-value problem admits a unique global smooth solution, provided that the initial data is smooth and small in…
An initial-boundary value problem with one boundary condition is considered for the higher order nonlinear Schr\"odinger equation. It is assumed that either the boundary condition is homogeneous or the nonlinearity in the equation is…