(Un)Stable Manifold Computation via Iterative Forward-Backward Runge-Kutta Type Methods

I present numerical methods for the computation of stable and unstable manifolds in autonomous dynamical systems. Through differentiation of the Lyapunov-Perron operator in [Casteneda, Rosa 1996], we find that the stable and unstable manifolds are boundary value problems on the original set of differential equation. This allows us to create a forward-backward approach for manifold computation, where we iteratively integrate one set of variables forward in time, and one set of variables backward in time. Error and stability of these methods is discussed

Hawking Radiation and Classical Tunneling: a Numerical Study

Unruh [1] demonstrated that black holes have an analogy in acoustics. Under this analogy the acoustic event horizon is defined by the set of points in which the local background flow exceeds the local sound speed. In past work [2], we demonstrated that under a white noise source, the acoustic event horizon will radiate at a thermal spectrum via a classical tunneling process. In this work, I summarize the theory presented in [2] and nondimensionalize it in order to reduce the dynamical equations to one parameter, the coupling coefficient η2. Since η2 is the sole parameter of the system, we are able to vary it in a numerical study of the dependence of the transmission coefficient on η2. This numerical work leads to the same functional dependence of the transmission coefficient on η2 as predicted in [2]

Exact Coherent Structures in Fully Developed Two-Dimensional Turbulence

This paper reports several new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions have a number of remarkable properties which distinguish them from analogous solutions of the Navier-Stokes equation describing transitional flows. First of all, they come in high-dimensional continuous families. Second, solutions of different types are connected, e.g., an equilibrium can be smoothly continued to a traveling wave or a time-periodic state. Third, and most important, many of these solutions are dynamically relevant for turbulent flow at high Reynolds numbers. Specifically, we find that turbulence in numerical simulations exhibits large-scale coherent structures resembling some of our time-periodic solutions both frequently and over long temporal intervals. Such solutions are analogous to exact coherent structures originally introduced in the context of transitional flows

Hawking radiation and classical tunneling: A ray phase space approach

Acoustic waves in fluids undergoing the transition from sub-to supersonic flow satisfy governing equations similar to those for light waves in the immediate vicinity of a black hole event horizon. This acoustic analogy has been used by Unruh and others as a conceptual model for Hawking radiation. Here, we use variational methods, originally introduced by Brizard for the study of linearized MHD, and ray phase space methods, to analyze linearized acoustics in the presence of background flows. The variational formulation endows the evolution equations with natural Hermitian and symplectic structures that prove useful for later analysis. We derive a 2 x 2 normal form governing the wave evolution in the vicinity of the event horizon. This shows that the acoustic model can be reduced locally (in ray phase space) to a standard (scalar) tunneling process weakly coupled to a unidirectional non-dispersive wave (the incoming wave ). Given the normal form, the Hawking thermal spectrum can be derived by invoking standard tunneling theory, but only by ignoring the coupling to the incoming wave. Deriving the normal form requires a novel extension of the modular ray-based theory used previously to study tunneling and mode conversion in plasmas. We also discuss how ray phase space methods can be used to change representation, which brings the problem into a form where the wave functions are less singular than in the usual formulation, a fact that might prove useful in numerical studies. (C) 2016 AIP Publishing LLC