Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from sub-critical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram-- specifically, determination of rigorous Eckhaus-type stability conditions-- remains an interesting open problem.Comment: 12 pages, 20 figure

On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions

In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multi-periodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.Comment: 19 pages

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

Intervocalic consonant sequences in Korean

This paper reports the results of an instrumental phonetic study of intervocalic consonant sequences in Korean. The study explored a putative positional neutralization produced at the phonetics/phonology interface. It was designed to determine whether Korean intervocalic laryngeal consonants are phonetically distinct from geminates, plain consonants, or laryngeal consonants in consonant clusters. The results showed that the contrast between intervocalic tensed singletons and geminates was neutralized, and that both of these patterned with heterorganic consonant sequences rather than plain singletons. Moreover, we found that this neutralization persisted across (limited) variation in speaking rate, although intervocalic tense consonants were more compressible in faster speech than were post-consonantal tense consonants

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation

In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy, Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader's convenience, we include in an appendix the corresponding treatment of the Swift-Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.Comment: 78 pages, 11 figure

Chest Pain and Costochondritis Associated with Vitamin D Deficiency: A Report of Two Cases

Vitamin D is integral for bone health, and severe deficiency can cause rickets in children and osteomalacia in adults. Although osteomalacia can cause severe generalized bone pain, there are only a few case reports of chest pain associated with vitamin D deficiency. We describe 2 patients with chest pain that were initially worked up for cardiac etiologies but were eventually diagnosed with costochondritis and vitamin D deficiency. Vitamin D deficiency is known to cause hypertrophic costochondral junctions in children (“rachitic rosaries”) and sternal pain with adults diagnosed with osteomalacia. We propose that vitamin D deficiency may be related to the chest pain associated with costochondritis. In patients diagnosed with costochondritis, physicians should consider testing and treating for vitamin D deficiency

The Shape of LITTLE THINGS Dwarf Galaxies DDO 46 and DDO 168: Understanding the stellar and gas kinematics

We present the stellar and gas kinematics of DDO 46 and DDO 168 from the LITTLE THINGS survey and determine their respective Vmax/sigma_z,0 values. We used the KPNO's 4-meter telescope with the Echelle spectrograph as a long-slit spectrograph. We acquired spectra of DDO 168 along four position angles by placing the slit over the morphological major and minor axes and two intermediate position angles. However, due to poor weather conditions during our observing run for DDO 46, we were able to extract only one useful data point from the morphological major axis. We determined a central stellar velocity dispersion perpendicular to the disk, sigma_z,0, of 13.5+/-8 km/s for DDO 46 and of 10.7+/-2.9 km/s for DDO 168. We then derived the maximum rotation speed in both galaxies using the LITTLE THINGS HI data. We separated bulk motions from non-circular motions using a double Gaussian decomposition technique and applied a tilted-ring model to the bulk velocity field. We corrected the observed HI rotation speeds for asymmetric drift and found a maximum velocity, Vmax, of 77.4 +/- 3.7 and 67.4 +/- 4.0 km/s for DDO 46 and DDO 168, respectively. Thus, we derived a kinematic measure, Vmax/sigma_z,0, of 5.7 +/- 0.6 for DDO 46 and 6.3 +/- 0.3 for DDO 168. Comparing these values to ones determined for spiral galaxies, we find that DDO 46 and DDO 168 have Vmax/sigma_z,0 values indicative of thin disks, which is in contrast to minor-to-major axis ratio studies