We consider the pseudorelativistic Hartree equation

$$ \begin{align*} i\partial_t\psi=(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\psi-(|x|^{-1}*|\psi|^2)\psi\quad \text{with } (t,x)\in\mathbb{R}\times\mathbb{R}^3, \end{align*} $$

which describes the dynamics of pseudorelativistic boson stars in the mean-field limit. We study the travelling waves of the form $\psi (t,x)=e^{it\mu }\varphi _{c}(x-vt)$, where $v\in \mathbb {R}^3$ denotes the travelling velocity. We prove that $\varphi _{c}$ converges strongly to the minimiser $\varphi _{\infty }$ of the limit energy $E_{\infty }(N)$ in $H^1(\mathbb {R}^3)$ as the light speed $c\to \infty $, where $E_{\infty }(N)$ is the corresponding energy for the limit equation

$$ \begin{align*} -\frac{1}{2m}\Delta\varphi_{\infty}+i(v\cdot\nabla)\varphi_{\infty}-({|x|^{-1}}*|\varphi_{\infty}|^2)\varphi_{\infty}=-\lambda\varphi_{\infty}. \end{align*} $$

Since the operator $-\Delta $ is the classical kinetic operator, we call this the nonrelativistic limit. We prove the existence of the minimiser for the limit energy $E_{\infty }(N)$ by using concentration-compactness arguments.

Recently, we analysed spontaneous symmetry breaking (SSB) of solitons in linearly coupled dual-core waveguides with fractional diffraction and cubic nonlinearity. In a practical context, the system can serve as a model for optical waveguides with the fractional diffraction or Bose–Einstein condensate of particles with Lévy index $\alpha <2$. In an earlier study, the SSB in the fractional coupler was identified as the bifurcation of subcritical type, becoming extremely subcritical in the limit of $\alpha \rightarrow 1$. There, the moving solitons and collisions between them at low speeds were also explored. In the present paper, we present new numerical results for fast solitons demonstrating restoration of symmetry in post-collision dynamics.

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

The main result is that the ellipticity and the Fredholm property of a $\Psi $ DO acting on Sobolev spaces in the Weyl-Hörmander calculus are equivalent when the Hörmander metric is geodesically temperate and its associated Planck function vanishes at infinity. The proof is essentially related to the following result that we prove for geodesically temperate Hörmander metrics: If $\lambda \mapsto a_{\lambda }\in S(1,g)$ is a $\mathcal {C}^N$ , $0\leq N\leq \infty $ , map such that each $a_{\lambda }^w$ is invertible on $L^2$ , then the mapping $\lambda \mapsto b_{\lambda }\in S(1,g)$ , where $b_{\lambda }^w$ is the inverse of $a_{\lambda }^w$ , is again of class $\mathcal {C}^N$ . Additionally, assuming also the strong uncertainty principle for the metric, we obtain a Fedosov-Hörmander formula for the index of an elliptic operator. At the very end, we give an example to illustrate our main result.

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form

$$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$

$\tau :{\open R}^n\to {\open R}^n$

$\tau (x)=0$

$\tau (x)=x$

$\tau (x)={x}/{2}$

${\open R}^n$

This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type

$$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$

We investigate the decay for |x|→∞ of weak Sobolev-type solutions of semilinear nonlocal equations Pu = F(u). We consider the case when P = p(D) is an elliptic Fourier multiplier with polyhomogeneous symbol p(ξ), and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation

for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.

In this paper we prove weighted norm inequalities with weights in the ${{A}_{p}}$ classes, for pseudodifferential operators with symbols in the class $S_{\rho ,\delta }^{n(\rho -1)}$ that fall outside the scope of Calderón–Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy–Littlewood type maximal functions. Our weighted norm inequalities also yield ${{L}^{p}}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\text{OPS}_{\rho ,\delta }^{m}$.