12 research outputs found
Modified Scattering for the Hartree Nonlinear Schrödinger Equation
No full textWe prove sharp decay and modified scattering for the Hartree nonlinear Schrödinger equation in dimensions and using the testing by wavepackets method of Ifrim and Tataru. We show that the scattering behavior happens at a regularity well below that of earlier results of Hayashi-Naumkin and Kato-Pusateri.21 pages, comments welcome
Modified Scattering for the Schr\"odinger-Bopp-Podolsky Equation
No full textWe prove sharp decay and modified scattering for the
Schr\"odinger-Bopp-Podolsky equation in and spatial dimensions with
small initial data chosen from a weighted Sobolev space.Comment: 17 pages, comments welcome
Several problems in nonlinear Schrödinger equations
No full text“We study several different problems related to nonlinear Schrödinger equations….
We prove several new results for the first equation: a modified scattering result for both an averaged version of the equation and the full equation, as well as a set of Strichartz estimates and a blowup result for the 3d cubic problem.
We also present an exposition of the classical work of Bourgain on invariant measures for the second equation in the mass-subcritical regime”--Abstract, page iv
Modified scattering for a dispersion-managed nonlinear Schr\"odinger equation
No full textWe prove sharp decay and modified scattering for a one-dimensional
dispersion-managed cubic nonlinear Schr\"odinger equation with small initial
data chosen from a weighted Sobolev space. Specifically, we work with an
averaged version of the dispersion-managed NLS in the strong dispersion
management regime. The proof adapts techniques from Hayashi-Naumkin and
Kato-Pusateri, which established small-data modified scattering for the
standard cubic NLS.Comment: 9 page
Well-posedness and blowup for the dispersion-managed nonlinear Schr\"odinger equation
No full textWe consider the nonlinear Schr\"odinger equation with periodic dispersion
management. We first establish global-in-time Strichartz estimates for the
underlying linear equation with suitable dispersion maps. As an application, we
establish a small-data scattering result for the cubic equation. Finally,
we use a virial argument to demonstrate the existence of blowup solutions for
the cubic equation with piecewise constant dispersion map.Comment: 13 pages, 1 figur
WELL-POSEDNESS AND BLOWUP FOR THE DISPERSION-MANAGED NONLINEAR SCHRÖDINGER EQUATION
No full textWe consider the nonlinear Schrödinger equation with periodic dispersion management. We first establish global-in-time Strichartz estimates for the underlying linear equation with suitable dispersion maps. As an application, we establish a small-data scattering result for the 3d cubic equation. Finally, we use a virial argument to demonstrate the existence of blowup solutions for the 3d cubic equation with piecewise constant dispersion map
Averaging for the dispersion-managed NLS
No full textWe establish global-in-time averaging for the -critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in .13 page
Averaging For The 2 D Dispersion-managed NLS
No full textWe establish global-in-time averaging for the L2-critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in L2
Katherine Drexel: Mystery, mission, spirituality and sainthood
No full textKatherine Drexel (1858-1955), the founder of the sisters of the blessed. Sacrament, was canonized by Pope John Paul II on October 1, 2000. This thesis analyzes Drexel’s life and virtues to establish why she became a saint. The examination of Drexel’s life begins in Chapter 2, which discloses the family life of a wealthy Philadelphia debutante, who, nonetheless, learned charity and philanthropy from her banker father and her religious mother. Following the deaths of her parents, Drexel wanted to enter a Catholic convent to spend her life in prayer and contemplation. Chapter 3 details the process of her vocational discernment that was carried out over several years in an epistolary argument with her spiritual director, Bishop O'Connor of Omaha. While he first believed that her vocation was to remain a single woman dedicated to serving the poor through judicial disbursement of her large inheritance, he later decided that she should found a new order of missionary nuns dedicated to the needs of the Native- Americans and African-Americans. Chapter 3 details the difficulties she encountered in the establishing of her new order at a time when the United States was racially divided by both law and custom. Drexel's order grew slowly in the face of open hostility towards her mission, including that of the Ku Klux Klan, and it then declined following the upheavals that came in the wake of the Second Vatican Council. Chapter 4 addresses the spirituality that sustained Drexel throughout her long life. Her deep spirituality was both kenotic and Eucharistie, and it allowed her to face daunting challenges in the mission field. Chapter 5 analyzes why the pope chose to canonize Drexel and entails a study of the process of saint-making as it evolved over the centuries
