Related papers: Long-time dynamics of completely integrable Schr\"…

We consider the nonlinear Schrodinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which…

We consider the one dimensional focusing (cubic) Nonlinear Schr\"odinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth…

We prove that solutions of the toroidal Schr{\"o}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums…

We present doubly-periodic solutions of the infinitely extended nonlinear Schrodinger equation with an arbitrary number of higher-order terms and corresponding free real parameters. Solutions have one additional free variable parameter that…

We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to…

We study the long time dynamics of the Schr\"odinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schr\"odinger equation can or cannot concentrate on a given closed geodesic. As an application, we…

We begin a study of a multi-parameter family of Cauchy initial-value problems for the modified nonlinear Schr\"odinger equation, analyzing the solution in the semiclassical limit. We use the inverse scattering transform for this equation,…

A derivation of the time-dependent Schr\"odinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent…

This paper investigates the properties of trajectories in harmonic oscillator systems equipped with a point, absolutely continuous, or singular measure. As demonstrated in [30], infinite-dimensional linear flows of countable oscillator…

This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small…

In this paper, we prove a reducibility result for a relativistic Schr\"odinger equation on torus with time quasi-periodic unbounded perturbations of order 1/2, and finally conjugate the original equation to a time independent, 2\times 2…

For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to…

We study the long time behavior of the stochastic quantization equation. Extending recent results by Mourrat and Weber we first establish a strong non-linear dissipative bound that gives control of moments of solutions at all positive times…

We study positive bound states for the semiclassical stationary nonlinear Schr\"odinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows…

We consider a nonlinear Schr{\"o}dinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the…

In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the…

In this paper, we analyze the long-time dynamics of small solutions to the $1d$ cubic nonlinear Schr\"odinger equation (NLS) with a trapping potential. We show that every small solution will decompose into a small solitary wave and a…

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution…

We study the survival probability associated with a semi-classical matrix Shr\"odinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual…

The purpose of the present work is to expand substantially the type of control and estimation problems that can be addressed following the paradigm of Schr\"odinger bridges, by incorporating termination (killing) of stochastic flows.…