Related papers: Existence of a minimal non-scattering solution to …

We have derived the extended Korteweg-de Vries equation describing the long gravity waves without limitation to surface deviation. The only restriction to the surface deviation is connected with the stability condition for appropriate…

We study the dynamics of multi-component Bose gas described by the Vector Nonlinear Schr\"{o}dinger Equation (VNLS), aka the Vector Gross--Pitaevskii Equation (VGPE) . Through a Madelung transformation, the VNLS can be reduced to coupled…

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic…

The purpose of this paper is twofold. In the first part, we provide a new proof of the global existence of the solutions to the Vlasov-Maxwell system with a small initial distribution function. Our approach relies on vector field methods,…

In this paper we prove that the defocusing, $d$-dimensional mass critical nonlinear Schr{\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$ and $d \geq 3$. To do this, we will prove…

In this paper we consider the long time behavior of solutions to the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. For sufficiently small, smooth, decaying data we prove…

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type…

We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schr\"odinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data.…

In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schr\"odinger equation on waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in…

Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg--de~Vries and modified Korteweg--de~Vries equation (for the last one both in focusing and defocusing cases) are constructed. The method of the…

We consider the Cauchy problem for the Korteweg--de Vries equation with real initial data $q$ that is both $L^1$ and $L^2$ summable and supported on (0,\infty). Using the left reflection coefficient and Hankel operators on the Hardy space…

We study the $L^2$-supercritical generalized Korteweg-de Vries equation (gKdV) with nonlinearities $p>5$. While local well-posedness in $H^1$ is classical, the long-time dynamics in the supercritical regime remains largely unexplored beyond…

In this paper, we study the critical norm conjecture for the inter-critical nonlinear Schr{\"o}dinger equation with critical index $s_c$ satisfying $\frac{1}{2}<s_c<1$ when $d\geq 5$. Under the assumption of uniform boundedness of the…

We obtain the existence, nonexistence and multiplicity of positive solutions with prescribed mass for nonlinear Schr\"{o}dinger equations in bounded domains via a global bifurcation approach. The nonlinearities in this paper can be mass…

We consider the focusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation \[ iu_t + \Delta u = -|x|^{-b}|u|^{\alpha}u \] where $n \geq 3$, $0<b<\min(2, n/2)$, and $\alpha=(4-2b)/(n-2)$. We prove the global well-posedness and…

New reductions for the multicomponent modified Korteveg-de Vries (MMKdV) equations on the symmetric spaces of {\bf DIII}-type are derived using the approach based on the reduction group introduced by A.V. Mikhailov. The relevant inverse…

We develop the techniques of \cite{KS1} and \cite{ES1} in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schr\"odinger…

We consider a mass-critical system of nonlinear Sch\"{o}dinger equations \begin{align*} \begin{cases} i\partial_t u +\Delta u =\bar{u}v,\\ i\partial_t v +\kappa \Delta v =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*}…

We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing…

In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schr\"odinger system on $\mathbb{R}^2$ below the threshold in $L_x^2h^1(\mathbb{R}^2\times \mathbb{Z})$. We first establish…