A class of nonlinear,completely integrable abstract wave equations

IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u+Lu+u _H ² u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u) are relatively compact inD(L ^1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball B=(u ₀,u ₀ )D(L ^1/2)×H; L^1/2 u _{0 H} ² +u ² < for sufficiently small.     

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

G-quadruplex DNA and ligand interaction in living cells using NMR spectroscopy

Gathering structural information from biologically relevant molecules inside living cells has always been a challenging task. In this work, we have used multidimensional NMR spectroscopy to probe DNA G-quadruplexes inside living Xenopus laevis oocytes. Some of these structures can be found in key regions of chromosomes. G-quadruplexes are considered potential anticancer therapeutic targets and several lines of evidence indirectly point out roles in key biological processes, such as cell proliferation, genomic instability or replication initiation. However, direct demonstrations of the existence of G-quadruplexes in vivo are scarce. Using SOFAST-HMQC type spectra, we probed a tetramolecular G-quadruplex model made of d(TG₄T)₄ inside living Xenopus laevis oocytes. Our observations lead us to conclude that the quadruplex structure is formed within the cell and that the intracellular environment preferentially selects a conformation that most resembles the one found in vitro under KCl conditions. We also show for the first time that specific ligands targeting G-quadruplexes can be studied using high resolution NMR directly inside living cells, opening new avenues to study ligand binding discrimination under physiologically relevant conditions with atomic detail.     

Orbital stability of standing waves for some nonlinear Schrödinger equations

We present a general method which enables us to prove the orbital stability of some standing waves in nonlinear Schrödinger equations. For example, we treat the cases of nonlinear Schrödinger equations arising in laser beams, of time-dependent Hartree equations ....     

Sur la composition de l'arôme de Thé noir IV

17 substances were shown to be present in the aroma of black tea. This brings the total number of substances now identified to 187. Many of these new aromatic constituents are present only in traces. Nevertheless they possess an appreciable olfactive intensity whose flowery and/or fruity characteristics are found in many perfumed black teas such as: Kenya, Darjeeling, Ceylan, China and Oolong. The components were identified by GC/MS. and their presence in the aroma verified by GC.     

The Cauchy problem for the nonlinear Schrödinger equation in H1

We consider the initial value problem for the nonlinear Schrödinger equation in H¹(Rⁿ). We establish local existence and uniqueness for a wide class of subcritical nonlinearities. The proofs make use of a truncation argument, space-time integrability properties of the linear equation, anda priori estimates derived from the conservation of energy. In particular, we do not need any differentiability property of the nonlinearity with respect to x.Research supported by NSF grants DMS 8201639 and DMS 8703096.     

Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball

In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation

Existence of excited states for a nonlinear Dirac field

We prove the existence of infinitely many stationary states for the following nonlinear Dirac equation $$i\gamma ^\mu \partial _\mu \psi - m\psi + (\bar \psi \psi )\psi = 0.$$ Seeking for eigenfunctions splitted in spherical coordinates leads us to analyze a nonautonomous dynamical system inR ². The number of eigenfunctions is given by the number of intersections of the stable manifold of the origin with the curve of admissible datum. This proves the existence of infinitely many stationary states, ordered by the number of nodes of each component.