Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case

Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling-wave solutions of viscous systems of conservation laws for systems of generic type, removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions.Comment: Fixed minor typo

Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation

In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles, we are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.Comment: 26 pages. Sign error corrected in Lemma 3. Statement of main theorem corrected. Exposition updated and references added

IDPAL - Input Decoupled Partially Adiabatic Logic: Implementation and Examination

This thesis presents the experimental results of a four-phase IDPAL eight-input exclusive-OR gate. The following problems with IDPAL are addressed: multistage circuits malfunctioning, simulation convergence anomalies, and inferring input information through the power clock current. EPAD MOSFETs, which provide a low threshold voltage, are shown to be unsuccessful in correcting the malfunctioning behavior of multilayer circuits. A solution to multilayer IDPAL circuits malfunctioning, called IDPAL with discharge, is shown. The differences between simulation waveforms produced by LTspice and the experimental circuits recorded by a Tektronix’s Oscilloscope are investigated. IDPAL is implemented and analyzed using ALD MOSFETs for the following adiabatic families: 2N-2P, IDPAL, and IDPAL with discharge

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider's renormalization techniques do not appear to appl

Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law

In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spectral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate period. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed "metastability" of the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou

Convergence of Hill's Method for Nonselfadjoint Operators

This is the published version, also available here: http://dx.doi.org/10.1137/100809349.By the introduction of a generalized Evans function defined by an appropriate 2-modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices

Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev–Petviashvili Equation

This is the published version, also available here: http://dx.doi.org/10.1137/090770758.In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev–Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles, we are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteweg–de Vries and modified Korteweg–de Vries equations