Self trapping transition for a nonlinear impurity within a linear chain

In the present work we revisit the issue of the self-trapping dynamical transition at a nonlinear impurity embedded in an otherwise linear lattice. For our Schr\"odinger chain example, we present rigorous arguments that establish necessary conditions and corresponding parametric bounds for the transition between linear decay and nonlinear persistence of a defect mode. The proofs combine a contraction mapping approach applied in the fully dynamical problem in the case of a 3D-lattice, together with variational arguments for the derivation of parametric bounds for the creation of stationary states associated with the expected fate of the self-trapping dynamical transition. The results are relevant for both power law nonlinearities and saturable ones. The analytical results are corroborated by numerical computations.Comment: 16 pages, 7 figures. To be published in Journal of Mathematical Physic

The probabilistic scaling paradigm

In this note we further discuss the probabilistic scaling introduced by the authors in [21, 22]. In particular we do a case study comparing the stochastic heat equation, the nonlinear wave equation and the nonlinear Schrodinger equation.Comment: Expository paper, 14 page

Corrigendum to: The TianQin project: current progress on science and technology

In the originally published version, this manuscript included an error related to indicating the corresponding author within the author list. This has now been corrected online to reflect the fact that author Jun Luo is the corresponding author of the article

WELL-POSEDNESS FOR THE CUBIC NONLINEAR SCHRÖDINGER EQUATIONS ON TORI

This thesis studies the cubic nonlinear Sch\ rodinger equation (NLS) on tori both from the deterministic and probabilistic viewpoints. In Part I of this thesis, we prove global-in-time well-posedness of the Cauchy initial value problem for the defocusing cubic NLS on 4-dimensional tori and with initial data in the energy-critical space $H^1$. Furthermore, in the focusing case we prove that if a maximal-lifespan solution of the cubic NLS \, $u: I\times\mathbb{T}^4\to \mathbb{C}$\, satisfies $\sup_{t\in I}\|u(t)\|_{\dot{H}^1(\mathbb{T}^4)