SEMINARIO DOBLE DE ECUACIONES EN DERIVADAS PARCIALES - Traveling waves for the nonlocal Gross-Pitaevskii equation // Local and nonlocal ODEs in the singular fractional Yamabe problem
Speaker: Salvador López-Martínez, Universidad Autónoma de Madrid, y Hardy Chan, ICMAT, ()Date: Friday, 19 November 2021 - 12:30Place: Aula 520, Dpto. de Matemáticas (UAM) y online - us06web.zoom.us/j/84300910588?pwd=ZjMvdDhNclhlRUVIMitPOGFGckhpUT09
"Traveling waves for the nonlocal Gross-Pitaevskii equation", Salvador López-Martínez (Universidad Autónoma de Madrid)
Abstract: In their seminal works, Gross and Pitaevskii proposed a Schrödinger type PDE to model a Bose gas. The nonlinearity in the equation presents a convolution with a potential that represents the interaction between the particles in the gas. Thus, the equation is nonlocal unless the potential is a Dirac delta, which corresponds to a repulsive contact interaction. This is a typical approximation and the problem in this local case has been extensively studied. In the talk we will consider more general interactions and present some new results, in collaboration with André de Laire, on the existence and properties of "dark solitons", which have been observed for instance in Bose-Einstein condensates. In our context, dark solitons are finite-energy traveling wave solutions that do not vanish at infinity. The proof of the existence is performed in two stages: first, we prove existence for almost every subsonic wave speed via a Mountain-Pass argument, and second, we fill the whole range of speeds for a family of potentials that provide suitable a priori estimates. We show also some properties of the solitons such as exponential decay or analyticity. We will finish by discussing a few open problems.
"Local and nonlocal ODEs in the singular fractional Yamabe problem", Hardy Chan (ICMAT)
Abstract: In conformal geometry, the Yamabe problem asks for Yamabe metrics, or conformal metrics of constant scalar curvature. In search of singular Yamabe metrics, one is led to the study of the Lane-Emden equation with a Sobolev-subcritical exponent that depends on the dimension of the singularity. The radial profile, which solves a classical ODE, is well-understood.
One could pose the same problem concerning the fractional curvature, a general notion that includes the scalar curvature, the curvatures associated to Paneitz and GJMS operators, as well as those with non-integer order. For the investigation of the corresponding radial profile, we discuss the development of the nonlocal ODE theory. Apart from the localizing Caffarelli-Silvestre extension, we show that nonlocal ODE can also be understood as a coupled infinite system of second order ODEs. Finally, we also mention a simple while surprising transformation that reduces the nonlocal ODE into almost a scalar first order ODE.