Differential Equations

Eigenvalue splitting of polynomial order for a system of Schrödinger operators with energy-level crossing.

Event Date: Jan 12, 2021 in Differential Equations, Seminars

In this talk I will present recent results on the spectral analysis of a 1D semiclassical system of coupled Schrödinger operators each of which has a simple well potential. Such systems arise as important models in the Born-Oppenheimer approximation of molecular dynamics. Focusing on the cases where the two underlying classical periodic trajectories cross to each other which imply an energy-level crossing, we give Bohr-Sommerfeld type quantization rules for the eigenvalues of the system in both tangential and transversal crossing cases. Our main results consists in the eigenvalue splitting...

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Solitons and multilinear Harmonic Analysis with potentials.

Event Date: Dec 09, 2020 in Differential Equations, Seminars

  Abstract: We study some classes of nonlinear dispersive PDEs with potentials in both 1 and 3 dimensions, motivated by questions on the stability of solitons and topological solitons. Our approach is based on the so-called “distorted Fourier transform” adapted to Schrodinger operators, and the development of multilinear harmonic analysis in this setting. This approach allows us to treat low power nonlinearities and describe the global space-time behavior of solutions, also by capturing singularities in frequency space which may arise due to various coherent phenomena. Several...

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Elapsed time neural assemblies and learning processes for weak interconnections.

Event Date: Dec 01, 2020 in Differential Equations, Seminars

Abstract: Modeling neural networks is an interesting problem from both mathematical and neuroscience point of view. In particular, evolution equations describing neural assemblies derived from stochastic processes and microscopic models have become a very active area in recent years. In this context we study a neural network via time-elapsed dynamics with learning, where neurons are described by their position in brain cortex and the elapsed time since last discharge. The learning process is modeled via the change in the interconnections among neurons which defines a learning rule. A typical...

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Strongly interacting multi-solitons and asymptotically static nonlinear waves.

Event Date: Nov 30, 1999 in Differential Equations, Seminars

Abstract: A generic global solution to a nonlinear wave equation exhibits oscillatory behavior, which is reflected in the fact that its kinetic energy does not tend to zero in infinite time. However, there are special solutions whose kinetic energy converges to zero and we call such solutions asymptotically stationary. These play an important role in the description of the phase portrait, whether they are stable or not. Familiar examples include stationary solutions (corresponding to critical points of the potential energy) or their stable manifolds. In this talk, we discuss asymptotically...

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Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities.

Event Date: Nov 10, 2020 in Differential Equations, Seminars

Abstract: The problem of the asymptotic stability of kinks in classical nonlinear scalar field equations on the real line leads to the study of the decay of small solutions to 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities. I will discuss the occurrence of a novel modified scattering behavior of such solutions that involves a logarithmic slow-down of the decay rate along certain rays. It is caused by a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. This talk is based...

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Gevrey regularity for the Vlasov-Poisson system.

Event Date: Nov 03, 2020 in Differential Equations, Seminars

Abstract: The importance of understanding high regularity solutions for the Vlasov-Poisson system has been underscored by the work on Landau damping of Mouhot-Villani and several other follow ups. These works on Landau damping make use of the propagation in time of high order regularity in a perturbative regime around homogeneous stationary solutions of the Vlasov-Poisson system. In this talk, we prove a general result on propagation of Gevrey regularity for the Vlasov-Poisson system on $\T^d\times \R^d$ using a Fourier space method in analogy to results proved for the 2D-Euler system by...

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