Related papers: Sharp local existence and nonlinear smoothing for …

We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the…

We prove that the cubic nonlinear Schr\"odinger equation (both focusing and defocusing) is globally well-posed in $H^s(\mathbb R)$ for any regularity $s>-\frac12$. Well-posedness has long been known for $s\geq 0$, see [55], but not…

In this paper, we study the probabilistic local well-posedness of the cubic Schr\"odinger equation (cubic NLS): \[ (i\partial_{t} + \Delta) u = \pm |u|^{2} u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener…

We prove a sharp local existence result for the Schr\"odinger-Korteweg-de Vries system with initial data in $H^k(\mathbb{R})\times H^s(\mathbb{R})$. The proof is based on the concept of \textit{integrated-by-parts strong solution}, which…

In this paper we study the generalized Korteweg de Vries (KdV) equation with the nonlinear term of order three: $(u^{3+1})_x$. We prove sharp local well--posedness for the initial and boundary value problem posed on the right half line. We…

We consider a two-dimensional nonlinear Schr\"odinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well…

In this paper, we study a class of one-dimensional nonlocal nonlinear Schr\"odinger equations on the line with nonlinearity given by a Fourier multiplier whose symbol has subcritical high-frequency growth. In terms of symbol order, this…

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations $$ \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,\ \ v(x,0)=\phi(x), $$ $$ \partial_tw + \alpha\partial_x^3w +…

This paper is concerned with the local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity. The equation arises as a non-integrable and lower nonlinearity version of the higher-order KdV…

In this paper we prove global well-posedness and scattering for the conformal, defocusing, nonlinear wave equation with radial initial data in the critical Sobolev space, for dimensions $d \geq 4$. This result extends a previous result…

We consider the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity. We establish the nonlinear smoothing properties of these equations, according to which the difference between the solution and the linear…

We establish that the quadratic non-linear Schr\"odinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and…

In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…

We consider two types of the generalized Korteweg - de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel…

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,&v(x,0)=\phi(x),\\ \partial_tw +…

We show that the 1d derivative nonlinear Schr\"{o}dinger equation (\ref{equ}) is globally well-posed in $H^s(\mathbb{R})$ for $s\geq 1/2$. We use the linear-nonlinear decomposition method to take advantage of the local smoothing effect of…

We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations…

We consider the $k$-dispersion generalized Benjamin-Ono ($k$-DGBO) equations. For nonlinearities with power $k \geq 4$, we establish local and global well-posedness results for the associated initial value problem (IVP) in both the critical…

In this paper, we prove a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type…

We study the Derivative Nonlinear Schr\"odinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and give a full…