The nonlinear Schrödinger equation with a potential

We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.     

Modified Scattering for the Boson Star Equation

We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior of such solutions at infinity by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, we also obtain global existence and (linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.     

Synthesis and evaluation of potent, highly-selective, 3-aryl-piperazinone inhibitors of protein geranylgeranyltransferase-I

A series of compounds based on the carboxyl-terminal CAAL sequence of PGGTase-I substrates was designed and synthesized. Using piperazin-2-one as a semi-rigid scaffold, we have introduced critical pharmacophores in a well-defined arrangement to mimic the CAAL sequence. High potency and exceptional selectivity were obtained for inhibition of PGGTase-I with structures such as 45 and 70. Potency of this series of GGTIs was dependent on the presence of an L-leucine residue with a free carboxyl terminus, as well as an S configuration of the 3-aryl group. The selectivity was significantly enhanced by 5-methyl substitution on the imidazole ring and fluorine substitution on the 3-aryl group. Modification of the 6-position of the piperazinone scaffold was found to be unfavorable. Compounds 44 and 69, the corresponding methyl esters of 45 and 70, were found to selectively block processing of Rap1A by PGGTase-I in whole cells with IC(50) values of 0.4 microM and 0.7 microM respectively.     

Scattering for the Zakharov System in 3 Dimensions

We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t ^?1, whereas the Schrödinger component decays almost at a rate of t ^?7/6.     

Global Analysis of a Model for Capillary Water Waves in Two Dimensions

In this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 (the capillary waves system). The model we consider here retains most of the difficulties of the full capillary water waves system, including the delicate time resonance structure and modified scattering. It is slightly simpler, however, at the technical level, and our goal here is to present our method in this simplified situation. The analysis of the full system is presented in a preprint by the authors (2014). © 2016 Wiley Periodicals, Inc.