Related papers: Scattering problem for the generalized Korteweg-de…
We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schr\"{o}dinger equation in weighted Sobolev space $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$. The key to prove this…
Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the…
In this study, we present a comprehensive global spectral analysis of the convection dispersion equation, which is also referred to in specific contexts as the Korteweg de Vries (KdV) equation, to investigate the behaviour of high order…
In this paper we prove that the defocusing, cubic nonlinear Schr{\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{2})$. To do this, we will prove a frequency localized interaction…
In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg--de Vries (KdV) equation in the homogeneous Sobolev spaces $\dot{H}^s$, for $s\ge0$. Specifically, we…
In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=\phi(x), \qquad 0<x<L, \ t>0 \qquad (1) \end{equation} subject…
We prove local-in-time a-priori estimates in $H^{-1}(\mathbb{R})$ for a family of generalized Korteweg--de Vries equations. This is the first estimate for any non-integrable perturbation of the KdV equation that matches the regularity of…
We study the scattering for the energy-subcritical stochastic nonlinear Schr\"odinger equation (SNLS) with additive noise. In particular, we examine the long-time behavior of solutions associated with the noise…
In this article we present a numerical analysis for a third-order differential equation with non-periodic boundary conditions and time-dependent coefficients, namely, the linear Korteweg-de Vries Burgers equation. This numerical analysis is…
The formal scattering theory is developed for the three-particle differential Faddeev equations. The theory is realised along the same line as in the standard two-body case. The solution of the scattering problem is expressed in terms of…
This work concerns the study of persistence property in polynomial weighted spaces for solutions of the generalized fractional KdV equation in any spatial dimension $d\geq 1$. By establishing well-posedness results in conjunction with some…
We consider the cubic nonlinear Schr\"odinger equation, posed on $\R^n\times M$, where $M$ is a compact Riemannian manifold and $n\geq 2$. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a…
In this paper we construct global strong dispersive solutions to the space inhomogeneous kinetic wave equation (KWE) which propagate $L^1_{xv}$ -- moments and conserve mass, momentum and energy. We prove that they scatter, and that the wave…
In this work we establish a dispersive blow-up result for the initial value problem (IVP) for the fifth order Korteweg-de Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{-2mm}=0,&\quad…
In this paper, we prove scattering for the defocusing Beam equation u_{tt}+D^2u+mu+ |u|^{p-1}u=0 in the energy space in low dimensions 1< n <5 for p>1+8/n. The main difficulty is the absence of a Morawetz-type estimate and of a Galilean…
We obtain almost-sure scattering for the cubic defocusing Schr{\"o}dinger equation in the Euclidean space {$\mathbb{R}^3$}, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness…
We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schr\"odinger equation with variable coefficients. Previous scattering results for such problems address only the cases with constant coefficients or assume…
In this paper we prove global well-posedness and scattering for the defocusing, cubic, nonlinear wave equation on $\mathbf{R}^{1 + 3}$ with radial initial data lying in the critical Sobolev space $\dot{H}^{1/2}(\mathbf{R}^{3}) \times…
Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials…
In this paper we prove that the focusing, $d$-dimensional mass critical nonlinear Schr{\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$, $\| u_{0} \|_{L^{2}(\mathbf{R}^{d})} < \| Q…