In this note, we study the scattering amplitude for the Schrödinger equation with constant magnetic field. We consider the case where the strengh of the magnetic field goes to infinity and we discuss the competition between the magnetic and the electrostatic effects.

We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions ${\chi }_a(t,x)$ form a family of evolving regular curves in ${ℝ}^3$ that develop a singularity in finite time, indexed by a parameter $a>0$. We consider curves that are small regular perturbations of ${\chi }_a(t_0,x)$ for a fixed time $t_0$. In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...

We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space ${\dot{B}}_2^{\frac{n}{2}-\frac{2}{p},\infty }\left({𝐑}^n\right)$, when the nonlinearity is of type $u^p$, for $p\in 𝐍$. This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.

In this paper, we study the semiclassical limit of the cubic nonlinear Schrödinger equation with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches $0.$

We consider a singularly perturbed elliptic equation ${ϵ}^2\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {ℝ}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions on $f$. In...

We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential $V$.

Small data scattering for nonlinear Schrödinger equations (NLS), nonlinear wave equations (NLW), nonlinear Klein-Gordon equations (NLKG) with power type nonlinearities is studied in the scheme of Sobolev spaces on the whole space ${ℝ}^n$ with order $s\<n/2$. The assumptions on the nonlinearities are described in terms of power behavior $p_1$ at zero and $p_2$ at infinity such as $1+4/n\le p_1\le p_2\le 1+4/(n-2s)$ for NLS and NLKG, and $1+4/(n-1)\le p_1\le p_2\le 1+4/(n-2s)$ for NLW.

In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space Hs. We prove the existence and uniqueness of global solutions for small data in Hs with s > 1...

We review some recent results concerning Gibbs measures for nonlinear Schrödinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.