Related papers: On a bilinear restriction estimate for Schr\"oding…
We study a filtered Lie splitting scheme for the cubic nonlinear Schr\"{o}dinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in $H^s$ with $0<s<1$ overcoming the…
In this paper, we study the local well-posedness of nonlinear Schr\"odinger equations on tori $\mathbb{T}^{d}$ at the critical regularity. We focus on cases where the nonlinearity $|u|^{a}u$ is non-algebraic with small $a>0$. We prove the…
For a nonlinear Schr\"odinger system with mass critical exponent, we prove the existence and orbital stability of standing-wave solutions obtained as minimizers of the underlying energy functional restricted to a double mass constraint. In…
We prove bilinear inequalities for differential operators in $\mathbb{R}^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However,…
Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with…
Our first purpose is to extend the results from \cite{T} on the radial defocusing NLS on the disc in $\mathbb{R}^2$ to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs…
New Strichartz estimates for the modulated cubic nonlinear Schr\"{o}dinger equation are proved. These Strichartz estimates allow us to show that this equation is pathwise locally well-posed. We also show that improved Strichartz estimates…
In this paper, we investigate the multilinear boundedness properties of the higher ($n$-th) order Calder\'on commutator for dimensions larger than two. We establish all multilinear endpoint estimates for the target space…
We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schr\"odinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the…
The two-dimensional cubic nonlinear Schr\"{o}dinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schr\"{o}dinger equations are…
The restriction problem is better understood for hypersurfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for…
We consider the following wave guide nonlinear Schr\"odinger equation, \begin{equation} (i\partial \_t+\partial \_{xx}-\vert D\_y\vert )U=\vert U\vert ^2U\ \tag{WS} \end{equation} on the spatial cylinder $\mathbb{R} \_x\times \mathbb{T}…
In this paper the classical and nonlocal semi-discrete nonlinear Schr\"{o}dinger (sdNLS) equations with nonzero backgrounds are solved by means of the bilinearization-reduction approach. In the first step of this approach, the unreduced…
As local and nonlocal reductions of a discrete second-order Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr\"odinger type equations are considered. Through the bilinearization reduction method, we construct double…
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…
We consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick…
The $X^{s,b}$ spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behaviour of non-linear dispersive equations. It is of particular interest to obtain…
This paper explores the cubic-quintic Schr\"odinger equation in the entire Euclidean space. Our objectives are twofold: first, to advance the understanding of unresolved issues related to this equation, which are well known in the…
We consider a two-component system of cubic nonlinear Schr\"odinger equations in one space dimension. We show that each component of the solutions to this system behaves like a free solution in the large time, but there is a strong…
In this paper, kinds of Schr\"{o}dinger type equations with slowly decaying linear potential and power type or convolution type nonlinearities are considered. By using the concentration compactness principle, the sharp Gagliardo-Nirenberg…