Related papers: Quasilinear Schr\"odinger Equations
In this paper we consider Schr\"odinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 \pi \Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(\omega t)[u]$ where…
We establish the Kato-type smoothing property, i.e., global-in-time smoothing estimates with homogeneous weights, for the Schr\"odinger equation on Riemannian symmetric spaces of non-compact type and general rank. These form a rich class of…
The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schr\"{o}dinger equations mainly the compactness of the support and its spatial localization. This question is very related with pure…
We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we…
In this paper we show that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schr\"odinger equation. From this we immediately…
We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…
We establish Schauder-type estimates for linear parabolic systems driven by variable-coefficient nonlocal pseudo-differential operators of order $s>0$. These estimates are formulated in critical time-weighted H\"older/Besov-type spaces and…
We consider the Schrodinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with…
The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{\"o}dinger equation posed on $\mathbb{R}^d$, in a supercritical regime. Firstly, we establish Bourgain type bilinear estimates…
In this note, we consider the ill-posedness issue for the cubic nonlinear Schr\"odinger equation. In particular, we prove norm inflation based at every initial condition in negative Sobolev spaces below or at the scaling critical…
The primitive equations in a 3D infinite layer domain are considered with linearly growing initial data in the horizontal direction, which illustrates the global atmospheric rotating or straining flows. On the boundaries, Dirichlet, Neumann…
We prove exact controllability for quasi-linear Hamiltonian Schr\"odinger equations on tori of dimension greater or equal then two. The result holds true for sufficiently small initial conditions satisfying natural minimal regularity…
The large-time behavior of solutions to the derivative nonlinear Schr\"{o}dinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our…
In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in…
We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which…
We prove two new results about the Cauchy problem for nonlinear Schroedinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinearities and includes approximations of cubic…
We consider semilinear Schr\"odinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value…
Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for…
We construct local solutions to the Benjamin-Ono equation for quasi-periodic initial data. The solution is unique among limits of smooth solutions and depends continuously on the data. Our result applies to a richer class of quasi-periodic…
Consider a nonlinear Schr\"odinger equation in $R^3$ whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in $H^1 \cap L^1(R^3)$ and inside a neighborhood of the first…