We consider radial solutions to the cubic Schrödinger equation on the Heisenberg group

$i{\partial }_{t}u-{\mathrm{\Delta }}_{{ℍ}^1}u={\left|u\right|}^{2}u,{\mathrm{\Delta }}_{{ℍ}^1}=\frac{1}{4}\left({\partial }_x^2+{\partial }_y^2\right)+\left(x^2+y^2\right){\partial }_s^2,\left(t,x,y,s\right)\in ℝ\times {ℍ}^1.$

This equation is a model for totally nondispersive evolution equations. We show existence of ground state traveling waves with speed $\beta \in \left(-1,1\right)$. When the speed $\beta$ is sufficiently close to $1$, we prove their uniqueness up to symmetries and their smoothness along the parameter $\beta$. The main ingredient is the emergence of a limiting system as $\beta$ tends to the limit $1$, for which we establish linear stability of the ground state traveling wave.

We consider Anosov flows on closed 3-manifolds preserving a volume form $\mathrm{\Omega }$. Following Dyatlov and Zworski (Invent. Math. 210:1 (2017), 211–229) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $\left[{\iota }_X\mathrm{\Omega }\right]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott–Ruelle resonance multiplicities under the assumption of semisimplicity. We prove various results regarding semisimplicity on 1-forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows. We also study non-null-homologous deformations of contact Anosov flows, and we show that there is always a splitting Pollicott–Ruelle resonance on 1-forms and that semisimplicity persists in this instance. These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in the resonant spaces and in the Ruelle zeta function.

We first prove semiclassical resolvent estimates for the Schrödinger operator in ${ℝ}^d$, $d\ge 3$, with real-valued potentials which are Hölder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in ${ℝ}^d$, $d\ge 2$, and real-valued potentials which are Hölder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be Hölder, Lipschitz or just $L^{\infty }$.

The resonances for the Wigner–von Neumann-type Hamiltonian are defined by the periodic complex distortion in the Fourier space. Also, following Zworski, we characterize resonances as the limit points of discrete eigenvalues of the Hamiltonian with a quadratic complex-absorbing potential in the viscosity-type limit.

We consider the minimization of the functional

$$J\left[u\right]:={\int }_{\mathrm{\Omega }}\left(|\mathrm{\Delta }u{|}^2+{\chi }_{\left\{u>0\right\}}\right)$$

in the admissible class of functions

$$\mathcal{𝒜}:=\left\{u\in W^{2,2}\left(\mathrm{\Omega }\right):u-u_0\in W_0^{1,2}\left(\mathrm{\Omega }\right)\right\}.$$

Here, $\mathrm{\Omega }$ is a smooth and bounded domain of ${ℝ}^n$ and $u_0\in W^{2,2}\left(\mathrm{\Omega }\right)$ is a given function defining the Navier type boundary condition.

When $n=2$, the functional $J$ can be interpreted as a sum of the linearized Willmore energy of the graph of $u$ and the area of $\left\{u>0\right\}$ on the $xy$-plane.

The regularity of a minimizer $u$ and that of the free boundary $\partial \left\{u>0\right\}$ are very complicated problems. The most intriguing part of this is to study the structure of $\partial \left\{u>0\right\}$ near singular points, where $\nabla u=0$ (of course, at the nonsingular free boundary points where $\nabla u\ne 0$, the free boundary is locally $C^1$ smooth).

The scale invariance of the problem suggests that, at the singular points of the free boundary, quadratic growth of $u$ is expected. We prove that $u$ is quadratically nondegenerate at the singular free boundary points using a refinement of Whitney’s cube decomposition, which applies, if, for instance, the set $\left\{u>0\right\}$ is a John domain.

The optimal growth is linked with the approximate symmetries of the free boundary. More precisely, if at small scales the free boundary can be approximated by zero level sets of a quadratic degree two homogeneous polynomial, then we say that $\partial \left\{u>0\right\}$ is rank-2 flat.

Using a dichotomy method for nonlinear free boundary problems, we also show that, at the free boundary points $x\in \mathrm{\Omega }$, where $\nabla u\left(x\right)=0$, the free boundary is either well approximated by zero sets of quadratic polynomials, i.e., $\partial \left\{u>0\right\}$ is rank-2 flat, or $u$ has quadratic growth.

More can be said if $n=2$, in which case we obtain a monotonicity formula and show that, at the singular points of the free boundary where the free boundary is not well approximated by level sets of quadratic polynomials, the blow-up of the minimizer is a homogeneous function of degree two.

In particular, if $n=2$ and $\left\{u>0\right\}$ is a John domain, then we get that the blow-up of the free boundary is a cone; and in the one-phase case, it follows that $\partial \left\{u>0\right\}$ possesses a tangent line in the measure theoretic sense.

Differently from the classical free boundary problems driven by the Laplacian operator, the one-phase minimizers present structural differences with respect to the minimizers, and one notion is not included into the other. In addition, one-phase minimizers arise from the combination of a volume type free boundary problem and an obstacle type problem, hence their growth condition is influenced in a nonstandard way by these two ingredients.

We extend the large-deviation results obtained by N. J. B. Aza and the present authors on atomic-scale conductivity theory of free lattice fermions in disordered media. Disorder is modeled by a random external potential, as in the celebrated Anderson model, and a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large-deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric current densities around their (classical) macroscopic value is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Here, the quantum fluctuations of linear response currents are shown to exist in the thermodynamic limit, and we mathematically prove that they are related to the rate function of the large-deviation principle associated with current densities. We also demonstrate that, in general, they do not vanish (in the thermodynamic limit), and the quantum uncertainty around the macroscopic current density disappears exponentially fast with an exponential rate proportional to the squared deviation of the current from its macroscopic value and the inverse current fluctuation, with respect to growing space (volume) scales.