We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, their velocity. We illustrate these results with numerical simulations. ∞ −∞ |ψ| 2ψ ψ x dx.

Abstract. Metastability is a common obstacle to performing long molecular dynamics simulations. Many numerical methods have been proposed to overcome it. One method is parallel replica dynamics, which relies on the rapid convergence of the underlying stochastic process to a quasi-stationary distribution. Two requirements for applying parallel replica dynamics are knowledge of the time scale on which the process converges to the quasi-stationary distribution and a mechanism for generating samples from this distribution. By combining a Fleming-Viot particle system with convergence diagnostics to simultaneously identify when the process converges while also generating samples, we can address both points. This variation on the algorithm is illustrated with various numerical examples, including those with entropic barriers and the 2D Lennard-Jones cluster of seven atoms.

In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schrödinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though we focus on a proof of the 3d cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability.Source code for verification of our comptuations, and for further experimentation, are available at

Abstract. In a variety of applications it is important to extract information from a probability measure μ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure ν, from within a simple class of measures, which approximates μ. This problem is studied in the case where the Kullback-Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where ν is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms. 1. Introduction. This paper is concerned with the problem of minimizing the Kullback-Leibler divergence between a pair of probability measures, viewed as a problem in the calculus of variations. We are given a measure μ, specified by its RadonNikodym derivative with respect to a reference measure μ 0 , and we find the closest element ν from a simpler set of probability measures. After an initial study of the problem in this abstract context, we specify to the situation where the reference measure μ 0 is Gaussian and the approximating set comprises Gaussians. It is necessarily the case that minimizers ν are then equivalent as measures to μ 0 , 1 and we use the Feldman-Hajek theorem to characterize such ν in terms of their inverse covariance operators. This induces a natural formulation of the problem as minimization over the mean, from the Cameron-Martin space of μ 0 , and over an operator from a weighted Hilbert-Schmidt space. We investigate this problem from the point of view of the calculus of variations, studying properties of minimizing sequences, regularization to improve the space in which operator convergence is obtained, and uniqueness under a slight strengthening of a log-convex assumption on the measure μ.In the situation where the minimization is over a convex set of measures ν, the problem is classical and completely understood [10]; in particular, there is uniqueness

Abstract. We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [10]. Our proofs follow the ideas by Martel [30] and Martel and Merle [35], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.