Related papers: Long time dynamics for the one dimensional non lin…
In this paper, we consider the hyperbolic nonlinear Schr\"odinger equations (HNLS) on $\mathbb{R}\times\mathbb{T}$. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for…
We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we…
In this paper, we extend $\overline\partial$ steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schr\"odinger (NNLS) equation with weighted Sobolev initial data %and finite density initial data \begin{align*}…
A fully non-linear kinetic Boltzmann equation for anyons and large initial data is studied in a periodic 1d setting. Strong L1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness, and…
In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schr\"odinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost…
We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schr\"odinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this…
The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we…
For certain non linear evolution equations, existence of global in time flows for large data is a fundamental and difficult question. In general, for dispersive and wave equations high regularity of the data does not automatically guarantee…
We consider the Cauchy problem of the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} =…
We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schr\"odinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with…
We consider the stochastic nonlinear Schr\"odinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness…
In this paper we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schr\"odinger equation in $\mathbb{R}^2$ with general data in the critical space $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$. We show that if a…
We consider the Cauchy problem of nonlinear Schr\"odinger equations (NLS) with almost periodic functions as initial data. We first prove that, given a frequency set $\pmb{\omega} =\{\omega_j\}_{j = 1}^\infty$, NLS is local well-posed in the…
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 $d\in\{3,4,5\}$…
Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider a quasi-periodically…
We consider a dispersive equation of Schr{\"o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY…
In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schr\"odinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some…
The Riemann-Hilbert approach is extended to discuss the well-posedness of the nonlinear Schr\"odinger-Gerdjikov-Ivanon equation. The Lipschitz continuity of potential in $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$ to scattering data is…
We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schr\"odinger equations in dimensions $n \geq 3$, for solutions which have large, but finite, energy and large, but finite,…
In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schr\"odinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and…