Related papers: On the periodic Schr\"odinger-Debye equation

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic case, the author proved…

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global…

In this work we establish a local existence theory for the initial value problem associated to the general quasi-linear ultrahyperbolic Schr\"odinger equation.

We prove global well-posedness for low regularity data for the $L^2-critical$ defocusing nonlinear Schr\"odinger equation (NLS) in 2d. More precisely we show that a global solution exists for initial data in the Sobolev space $H^{s}(\mathbb…

For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-\Delta)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data…

We consider the fourth-order Schr\"odinger equation $$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u=0, $$ where $\alpha>0,\mu=\pm1$ or $0$ and $\lambda\in\mathbb{C}$. Firstly, we prove local well-posedness in…

We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction…

We consider the derivative nonlinear Schr\"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we…

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…

We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schr\"odinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive…

The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger equation in $\R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({\Bbb R^{n}})$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and…

We study the initial-boundary value problem for the derivative nonlinear Schr\"odinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost…

In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in…

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that…

We establish local well-posedness for the hyperbolic nonlinear Schrodinger equation (HNLS) in the critical spaces. Following the approach of Killip and Visan, we derive scale-invariant Strichartz estimates for HNLS on both rational and…

Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The…

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…

This work investigates radial solutions for nonlinear fractional Schr\"odinger equations driven by multiplicative noise. Leveraging radial deterministic and stochastic Strichartz estimates, we establish local well-posedness in the…

In dimensions greater than or equal to 3, we prove that the Schroedinger map initial-value problem is globally well-posed for small data in the critical Besov space.

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.