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This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0, Comment: Theorem 1.1-1.3 are restated. The paper is accepted by Commun. Math. Sci

Physical framework introductory concepts for wave motions Ray methods for linear waves Ray methods for nonlinear hyperbolic waves Ray method for the propagation of discontinuities generalized wavefront expansion for weak shocks small time analysis and shock stability Ray methods for nonlinear dispersive waves Ray methods for nonlinear dissipative waves interaction of dispersive waves interaction of hyperbolic waves.

We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of surface quasi-geostrophic (SQG) fronts with small slopes., 18 pages, 1 figure

We use contour dynamics to derive equations of motion for infinite planar surface quasi-geostrophic (SQG) fronts, and show that it leads to the same result as a regularization procedure introduced previously by Hunter and Shu (2018)., Comment: 12 pages, 2 figures

We derive a degenerate quasilinear Schr\"odinger equation that describes the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave.

The Burgers–Hilbert equation consists of an inviscid Burgers equation with a linear Hilbert-transform source term. We explain how the equation arises as a model for waves on a vorticity discontinuity and surface waves with constant frequency. We survey various results about the Burger–Hilbert equation, including ones on singularity formation, shock structure, weak solutions, and the enhanced life span of small, smooth solutions.

We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (GSQG) equations. We derive a cubic approximation of the contour dynamics equation, and prove the short-time well-posedness of the approximate equations for generalized surface quasi-geostrophic fronts and weak well-posedness for surface quasi-geostrophic fronts.

We derive a quadratically nonlinear equation that describes the motion of a tangential discontinuity in incompressible magnetohydrodynamics near the onset of a Kelvin–Helmholtz instability. We show that nonlinear effects enhance the instability and give a criterion for it to occur in terms of a longitudinal strain of the fluid motion along the discontinuity.

We use an asymptotic expansion introduced by Benilov and Pelinovskiĭ to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.

We derive an asymptotic equation that describes the propagation of weakly nonlinear surface waves on a tangential discontinuity in incompressible magnetohydrodynamics. The equation is similar to, but simpler than, previously derived asymptotic equations for weakly nonlinear Rayleigh waves in elasticity, and is identical to a model equation for nonlinear Rayleigh waves proposed by Hamilton et al. The most interesting feature of the surface waves is that their nonlinear self-interaction is nonlocal. As a result of this nonlocal nonlinearity, smooth solutions break down in finite time, and appear to form cusps.

We use a lubrication theory approximation to formulate a model for the reactive spreading of drops that deposit an autophobic monolayer of surfactant on a surface. The model consists of a Poisson equation on a moving domain with boundary conditions that depend on the history of the domain motion. We develop a numerical algorithm for solving the model, using the immersed interface method and the level-set method. Numerical solutions for traveling drops are qualitatively similar to experimental observations of reactive autophobic spreading.

We use a Hamiltonian formalism to derive equations for weakly nonlinear scale-invariant waves. We apply the results to nonlinear surface waves in elasticity and magnetohydrodynamics that satisfy nonlocal generalizations of the inviscid Burgers' equation.

We present a numerical solution of the Euler equations of gas dynamics for a weak-shock Mach reflection in a half-space. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agreement with the numerical solution of the unsteady transonic small-disturbance equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weak-shock Mach reflection. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.

An asymptotic analysis of the regular and Mach reflection of weak shocks leads to shock reflection problems for the unsteady transonic small disturbance equation. Numerical solutions of this equation resolve the von Neumann triple point paradox for weak shock Mach reflection. Related equations describe steady transonic shock reflections, weak shock focusing, and nonlinear hyperbolic waves at caustics.

We show that the resonant interaction of sound waves is a source of gravitational waves in general relativity. We derive an asymptotic equation that describes this process, and compute the interaction coefficient for the quadratically nonlinear coupling between sound waves and gravitational waves.

UV-B radiation suppresses cell-mediated immunity. Histidine forms trans-urocanic acid (trans-UCA) enzymatically in the stratum corneum. Photoisomerization of trans-UCA to cis-urocanic acid (cis-UCA) has been proposed for the initiation of an immunosuppressive process. Many microorganisms described in the literature metabolize histidine and/or trans-UCA. Our enrichment cultures of soil and sewage contain organisms that can degrade cis-UCA. We have tested microorganisms for degradation of cis-UCA, trans-UCA, or l -histidine when they are incorporated at 0.2% in nutrient broth. Six out of 10 selected genera isolated by our clinical microbiology laboratory degrade one or more of the imidazole substrates. We have cultured over 60 aerobic isolates from human skin. Of these, 33 degrade one or more of the three imidazole substrates and 12 degrade cis-UCA. Isolates from BALB/c mice are also active on cis-UCA. We have identified a cis-UCA-degrading bacterium as Micrococcus luteus. Four ATCC strains of M. luteus have been tested and three are active on histidine or trans-UCA; two are active on cis-UCA. Micrococci that degrade cis-UCA contain a new enzyme, cis-UCA isomerase, which converts the substrate to the trans-isomer. This enzyme provides access to the classical l -histidine degradation pathway. We hypothesize that an epidermal microflora that degrades l -histidine, trans-UCA, or cis-UCA influences the concentration of urocanic acids on the skin and, thus, affects immune suppression.

In this paper, we apply the Majda-Rosales theory of resonantly interacting, weakly nonlinear hyperbolic waves in one space dimension to the equations of magnetohydrodynamics. We write out the general system of resonant interaction equations for magnetohydrodynamics together with normalized triad equations for all possible magnetohydrodynamic triads. We consider both the strictly and the nonstrictly hyperbolic cases and we include both dissipative and dispersive effects. We present some numerical solutions of magnetosonic entropy wave interactions which illustrate the effect of KdV-type dispersion.

We study a nonlocal equation, analogous to the Kuramoto-Sivashinsky equation, in which short waves are stabilized by a possibly fractional diffusion of order less than or equal to two, and long waves are destabilized by a backward fractional diffusion of lower order. We prove the global existence, uniqueness, and analyticity of solutions of the nonlocal equation and the existence of a compact attractor. Numerical results show that the equation has chaotic solutions whose spatial structure consists of interacting traveling waves resembling viscous shock profiles.

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard-Burq-Zuily \cite{abz}, respectively by Wu \cite{wu} using different coordinates and methods. Instead our goal is improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained., Comment: 72 pages. We completely removed subsection 2.4. As a consequence, the proofs in Section 6 are much simpler and straightforward. Minor typos have been fixed

This paper derives a variable coefficient, multidimensional Burgers equation which models the propagation of a nonlinear sound wave through an incompressible background flow. The equation is derived from the compressible Euler equations by the combination of a weakly nonlinear acoustics expansion for the sound wave and an incompressible expansion for the background flow. The main effect of the incompressible flow on the sound wave is the advection of the sound wave by the transverse velocity component of the flow.

We study the nonlinear hyperbolic partial differential equation, (u t+uux)x=1/2u x 2 . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal. Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up. There are no a priori estimates on the second derivatives in any L p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation. We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.

We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (u t+uux)x=1/2u x 2 with the simplest initial data such that u x blows up in finite time. We prove that the zero-viscosity limit selects a unique global weak solution of the partial differential equation without viscosity. We also present numerical experiments which indicate that the zero-dispersion limit selects a different global weak solution of the same initial-value problem.

Urocanic acid (UCA) is a chromophore in the stratum corneum. Ultraviolet radiation (ultraviolet B) has been shown to suppress mammalian cell-mediated immunity. The photoisomerization of trans-UCA to cis-UCA was proposed as the initiator of the suppression process. Cis-urocanic acid has been demonstrated to suppress immunity by a variety of experiments. Investigators should be aware that laboratory illumination may be capable of interconverting trans-UCA and cis-UCA during experimental manipulations. This possible inadvertent contamination of one isomer by the other may influence results. We demonstrated that fluorescent lamps, daylight, sunlight and incandescent lamps were able to bring about isomerization. Window glass and container materials of plastic and clear glass did not filter out effective wavelengths, but three commercial plastic diffusers on fluorescent fixtures prevented the isomerization. Because the molar extinction coefficient (epsilon) for cis-UCA is less than that of trans-UCA, we have exposed 0.1 mM trans-UCA to ambient light and monitored the change in absorbance. A method is given to calculate the percentage of trans and cis isomers from the absorbance at 277 nm when the initial purity and absorbance are known. Using this procedure, we validated the molar extinction coefficient of cis-UCA.

We describe examples of hyperbolic surface waves and discuss their connection with initial boundary value and discontinuity problems for hyperbolic systems of PDEs that are weakly but not uniformly stable.

We study shock reflection for the two 2D Burgers equation. This model equation is an asymptotic limit of the Euler equations, and retains many of the features of the full equations. A von Neumann type analysis shows that the 2D Burgers equation has detachment, sonic, and Crocco points in complete analogy with gas dynamics. Numerical solutions support the detachment/sonic criterion for transition from regular to Mach reflection. There is also strong numerical evidence that the reflected shock in the 2D Burgers Mach reflection forms a smooth wave near the Mach stem, as proposed by Colella and Henderson in their study of the Euler equations.

We derive canonical asymptotic equations for weakly nonlinear solutions of conservation laws of mixed type. When two real wave speeds coalesce and become complex, we obtain the transonic small disturbance equation which changes type from hyperbolic to elliptic. When the coefficient of the nonlinear term in the transonic small disturbance equation vanishes, we derive cubically nonlinear 2 × 2 asymptotic equations. These include canonical equations which are parabolic on a line in state space and strictly hyperbolic away from this line.

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi-linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two-dimensional inviscid, incompressible fluid flows. Thus, the Burgers-Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers-Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near-identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers-Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small-amplitude smooth solutions of the Burgers-Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.

An asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed. The equation is used to study a nonlinear instability in the director field of a nematic liquid crystal. It is shown that smooth solutions of the asymptotic equation break down in finite time. Also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up.

The photomodulation of enzymes involves the activation and inactivation of enzyme reactions by UV and visible light. Enzymes or their reactions may be affected directly or indirectly. Direct effects involve photoproduction of a substrate, photodissociation of an inhibitor, photochemistry of protein amino acids, irradiation of a chromophore and irradiation of an enzyme substrate. Indirect effects involve gene expression, phytochrome and other photoreceptors which are not part of the enzyme, protein synthesis, membranes and photosynthesis. Photoactivation of enzymes is related to photocarcinogenesis, photomorphogenesis of plants, primary effects or side effects of phototherapy, deoxyribose nucleic acid (DNA) repair and many other aspects of biology and medicine. Model systems may contribute to the knowledge of protein chemistry and medicinal chemistry.

We use weakly nonlinear asymptotics to derive a canonical asymptotic equation for rotationally invariant hyperbolic waves. The equation can include weak dissipative, dispersive, or diffractive effects. We give applications to equations from magnetohydrodynamics, elasticity, and viscoelasticity.