8 research outputs found
Weakly coupled heat bath models for Gibbs-like invariant states in nonlinear wave equations
textabstractThermal bath coupling mechanisms as utilized in molecular dynamics are applied to partial differential equation models. Working from a semi-discrete (Fourier mode) formulation for the BurgersâHopf or Kortewegâde Vries equation, we introduce auxiliary variables and stochastic perturbations in order to drive the system to sample a target ensemble which may be a Gibbs state or, more generally, any smooth distribution defined on a constraint manifold. We examine the ergodicity of approaches based on coupling of the heat bath to the high wave numbers, with the goal of controlling the ensemble through the fast modes. We also examine different thermostat methods in the extent to which dynamical properties are corrupted in order to accurately compute the average of a desired observable with respect to the invariant distribution. The principal observation of this paper is that convergence to the invariant distribution can be achieved by thermostatting just the highest wave number, while the evolution of the slowest modes is little affected by such a thermostat
Discrete breathers in and related models
We touch upon the wide topic of discrete breather formation with a special
emphasis on the the model. We start by introducing the model and
discussing some of the application areas/motivational aspects of exploring time
periodic, spatially localized structures, such as the discrete breathers. Our
main emphasis is on the existence, and especially on the stability features of
such solutions. We explore their spectral stability numerically, as well as in
special limits (such as the vicinity of the so-called anti-continuum limit of
vanishing coupling) analytically. We also provide and explore a simple, yet
powerful stability criterion involving the sign of the derivative of the energy
vs. frequency dependence of such solutions. We then turn our attention to
nonlinear stability, bringing forth the importance of a topological notion,
namely the Krein signature. Furthermore, we briefly touch upon linearly and
nonlinearly unstable dynamics of such states. Some special aspects/extensions
of such structures are only touched upon, including moving breathers and
dissipative variations of the model and some possibilities for future work are
highlighted