Ricerc@Sapienza

Third order nonlinear evolution equations, that is the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV) equation and other ones are considered: they all are connected via Bäcklund transformations. These links can be depicted in a wide Bäcklund Chart which further extends the previous one constructed in [22]. In particular, the Bäcklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Bäcklund Chart.

We consider a system of differential equations of Monge–Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton–Jacobi equations on networks introduced in [P.-L. Lions and P. E. Souganidis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), pp. 535–545], we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.

We prove a representation formula of Hopf-Lax type for solutions to Hamilton-Jacobi equation involving a Caputo time-fractional derivative. Equations of this type are associated with optimal control problems where the controlled dynamics is given by a time-changed stochastic process describing the trajectory of a particle subject to random trapping effects.

Here we investigated the SH generation at the wavelength of 400 nm (pump laser at 800 nm, 120 fs pulses) of a "metasurface" composed by an alternation of GaAs nano-grooves and Au nanowires capping portions of flat GaAs. The nano-grooves depth and the Au nanowires thickness gradually vary across the sample. The samples are obtained by ion bombardment at glancing angle on a 150 nm Au mask evaporated on a GaAs plane wafer. The irradiation process erodes anisotropically the surface, creating Au nanowires and, at high ion dose, grooves in the underlying GaAs substrate (pattern transfer).

We consider an anisotropic hyperbolic equation with memory term: ?t2u(x,t)=?i,j=1n?i(aij(x)?ju)+?0t?|?|?2b?(x,t,?)?x?u(x,?)d?+R(x,t)f(x) for $x \in \Omega$ and $t\in (0, T)$ , which is a simplified model equation for viscoelasticity. The main result is a both-sided Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor $f(x)$ of the force term $R(x, t)\,f(x)$ .

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4and M=5are also given.

We built sign-changing solutions for a linear perturbation of the classical Yamabe problem.

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $??(0, 1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices.

We construct families of positive solutions for competitive and cooperative systems
which blow-up and concentrate at different points of the domain.
This problem can be seen as a generalization for systems of a Brezis–
Nirenberg type problem.

We consider the Cauchy problem for two prototypes of flux-saturated diffusion equations. In arbitrary space dimension, we give an optimal condition on the growth of the initial datum which discriminates between occurrence or nonoccurrence of a waiting time phenomenon. We also prove optimal upper bounds on the waiting time. Our argument is based on the introduction of suitable families of subsolutions and on a comparison result for a general class of flux-saturated diffusion equations.