Global Well-posedness for the Biharmonic Quintic Nonlinear Schr\"odinger Equation on R2\mathbb{R}^2

We prove that the Cauchy problem for the 2D quintic defocusing biharmonic Schr\"odinger equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $\frac{8}{7}<s<2$. Our main ingredient to establish the result is the $I$-method of Colliander-Keel-Staffilani-Takaoka-Tao \cite{colliander2002almost} which is used to construct the modified energy functional that is conserved in time

Multiplicity and concentration results for local and fractional NLS equations with critical growth

Goal of this paper is to study the following singularly perturbed nonlinear Schrödinger equation: eps^2s (-Delta)^s v + V(x)v = f(v), x in R^N, where s is in (0,1), N is greater or equal to 2, V in C(R^N,R) is a positive potential and f is assumed critical and satisfying general Berestycki-Lions type conditions. When eps is greater than 0 is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of V; in particular, we improve the result in [37]. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. Finally, we prove the previous results also in the limiting local setting s = 1 and N greater or equal to 3, with an exponential decay of the solutions

Light bullets in the spatiotemporal nonlinear Schrodinger equation with a variable negative diffraction coefficient

We report approximate analytical solutions to the (3+1)-dimensional spatiotemporal nonlinear Schr\"odinger equation, with the uniform self-focusing nonlinearity and a variable negative radial diffraction coefficient, in the form of three-dimensional solitons. The model may be realized in artificial optical media, such as left-handed materials and photonic crystals, with the anomalous sign of the group-velocity dispersion (GVD). The same setting may be realized through the interplay of the self-defocusing nonlinearity, normal GVD, and positive variable diffraction. The Hartree approximation is utilized to achieve a suitable separation of variables in the model. Then, an inverse procedure is introduced, with the aim to select a suitable profile of the modulated diffraction coefficient supporting desirable soliton solutions (such as dromions, single- and multilayer rings, and multisoliton clusters). The validity of the analytical approximation and stability of the solutions is tested by means of direct simulations

Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

In this paper we study the following nonlinear fractional Choquard-Pekar equation \begin{equation}\label{eq_abstract} (-\Delta)^s u + \mu u =(I_\alpha*F(u)) F'(u) \quad \hbox{in}\ \mathbb{R}^N, \tag{$*$} \end{equation} where $\mu>0$, $s \in (0,1)$, $N \geq 2$, $\alpha \in (0,N)$, $I_\alpha \sim \frac{1}{|x|^{N-\alpha}}$ is the Riesz potential, and $F$ is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions $u \in H^s(\mathbb{R}^N)$, by assuming $F$ odd or even: we consider both the case $\mu>0$ fixed and the case $\int_{\mathbb{R}^N} u^2 =m>0$ prescribed. Here we also simplify some arguments developed for $s=1$ in [Calc. Var. PDEs, 2022]. A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions [ARMA, 1983]; for \eqref{eq_abstract} the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as $\mu$ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any $m>0$. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a $C^1$-regularity.Comment: Advanced Nonlinear Studies (in press

On fractional Schrodinger equations with Hartree type nonlinearities

Goal of this paper is to study the following doubly nonlocal equation in the case of general nonlinearities of Berestycki-Lions type. We prove existence of ground states and we obtain regularity and asymptotic decay of general solutions, extending some results by Moroz and Van Schaftingen

On fractional Schrodinger equations with Hartree type nonlinearities

Goal of this paper is to study the following doubly nonlocal equation in the case of general nonlinearities of Berestycki-Lions type. We prove existence of ground states and we obtain regularity and asymptotic decay of general solutions, extending some results by Moroz and Van Schaftingen

On some nonlinear and nonlocal effective equations in kinetic theory and nonlinear optics

This thesis deals with some nonlinear and nonlocal effective equations arising in kinetic theory and nonlinear optics. First, it is shown that the homogeneous non-cutoff Boltzmann equation for Maxwellian molecules enjoys strong smoothing properties: In the case of power-law type particle interactions, we prove the Gevrey smoothing conjecture. For Debye-Yukawa type interactions, an analogous smoothing effect is shown. In both cases, the smoothing is exactly what one would expect from an analogy to certain heat equations of the form $\partial_t u = f(-\Delta)u$, with a suitable function $f$, which grows at infinity, depending on the interaction potential. The results presented work in arbitrary dimensions, including also the one-dimensional Kac-Boltzmann equation. In the second part we study the entropy decay of certain solutions of the Kac master equation, a probabilistic model of a gas of interacting particles. It is shown that for initial conditions corresponding to $N$ particles in a thermal equilibrium and $M\leq N$ particles out of equilibrium, the entropy relative to the thermal state decays exponentially to a fraction of the initial relative entropy, with a rate that is essentially independent of the number of particles. Finally, we investigate the existence of dispersion management solitons. Using variational techniques, we prove that there is a threshold for the existence of minimisers of a nonlocal variational problem, even with saturating nonlinearities, related to the dispersion management equation