Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure

Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude-frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and non-integrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, i.e., this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge

Stationary expansion shocks for a regularized Boussinesq system

Stationary expansion shocks have been recently identified as a new type of solution to hyperbolic conservation laws regularized by non-local dispersive terms that naturally arise in shallow-water theory. These expansion shocks were studied in [1] for the Benjamin-Bona-Mahony equation using matched asymptotic expansions. In this paper, we extend the analysis of [1] to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow water equations. The extension for a system is non-trivial, requiring a combination of small amplitude, long-wave expansions with high order matched asymptotics. The constructed asymptotic solution is shown to be in excellent agreement with accurate numerical simulations of the Boussinesq system for a range of appropriately smoothed Riemann data

Expansion shock waves in regularised shallow water theory

We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularised shallow water equations that include the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock’s existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock

Efficient Immortalization of luminal Epithelial Cells from Human Mammary gland by introduction of Simian virus 40 large Tumor antigen with a Recombinant Retrovirus

When defined in terms of markers for normal cell lineages, most invasive breast cancer cells correspond to the phenotype of the common luminal epithelial cell found in the terminal ductal lobular units. Luminal epithelial cells cultured from milk, which have limited proliferative potential, have now been immortalized by introducing the gene encoding simian virus 40 large tumor (T) antigen. Infection with a recombinant retrovirus proved to be 50-100 times more efficient than calcium phosphate transfection, and of the 17 cell lines isolated, only 5 passed through a crisis period as characterized by cessation of growth. When characterized by immunohistochemical staining with monoclonal antibodies, 14 lines showed features of luminal epithelial cells and of these, 7 resembled the common luminal epithelial cell type in the profile of keratins expressed. These cells express keratins 7, 8, 18, and 19 homogeneously and do not express keratin 14 or vimentin; a polymorphic epithelial mucin produced in vivo by luminal cells is expressed heterogeneously and the pattern of fibronectin staining is punctate. Although the cell lines have a reduced requirement for added growth factors, they do not grow in agar or produce tumors in the nude mouse. When the v-Ha-ras oncogene was introduced into two of the cell lines by using a recombinant retrovirus, most of the selected clones senesced, but one entered crisis and emerged after 3 months as a tumorigenic cell line

Dispersive Riemann problems for the Benjamin-Bona-Mahony equation

Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equation u t + u u x = u xxt are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation u t + u u x + u xxx = 0 . The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit as t → ∞ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem

Gamma-Ray Emissions from Pulsars: Spectra of the TEV Fluxes from Outer-Gap Accelerators

We study the gamma-ray emissions from an outer-magnetospheric potential gap around a rotating neutron star. Migratory electrons and positrons are accelerated by the electric field in the gap to radiate copious gamma-rays via curvature process. Some of these gamma-rays materialize as pairs by colliding with the X-rays in the gap, leading to a pair production cascade. Imposing the closure condition that a single pair produces one pair in the gap on average, we explicitly solve the strength of the acceleration field and demonstrate how the peak energy and the luminosity of the curvature-radiated, GeV photons depend on the strength of the surface blackbody and the power-law emissions. Some predictions on the GeV emission from twelve rotation-powered pulsars are presented. We further demonstrate that the expected pulsed TeV fluxes are consistent with their observational upper limits. An implication of high-energy pulse phase width versus pulsar age, spin, and magnetic moment is discussed.Comment: Revised to compute absolute TeV spectra (22 pages, 9 figures

Current Status of Musculoskeletal Trauma Care Systems Worldwide

BACKGROUND AND RATIONALE Although general trauma care systems and their effects on mortality reduction have been studied, little is known of the current state of musculoskeletal trauma delivery globally, particularly in low-income (LI) and low middle-income (LMI) countries. The goal of this study is to assess and describe the development and availability of musculoskeletal trauma care delivery worldwide. MATERIALS & METHODS A questionnaire was developed to evaluate different characteristics of general and musculoskeletal trauma care systems, including general aspects of systems, education, access to care and pre- and posthospital care. Surgical leaders involved with musculoskeletal trauma care were contacted to participate in the survey. RESULTS Of the 170 surveys sent, 95 were returned for use for the study. Nearly 30 percent of surgeons reported a formalized and coordinated trauma system in their countries. Estimates for the number of surgeons providing musculoskeletal trauma per one million inhabitants varied from 2.6 in LI countries to 58.8 in high-income countries. Worldwide, 15% of those caring for musculoskeletal trauma are fellowship trained. The survey results indicate a lack of implemented musculoskeletal trauma care guidelines across countries, with even high-income countries reporting less than 50% availability in most categories. Seventy-nine percent of the populations from LI countries were estimated to have no form of health care insurance. Formalized emergency medical services were reportedly available in only 33% and 50% of LI and LMI countries, respectively. Surgeons from LI and LMI countries responded that improvements in the availability of equipment (100%), number and locations of trauma-designated hospitals (90%), and physician training programs (88%) were necessary in their countries. The survey also revealed a general lack of resources for postoperative and rehabilitation care, irrespective of the country's income level. CONCLUSION This study addresses the current state of musculoskeletal trauma care delivery worldwide. These results indicate a greater need for trauma system development and support, from prehospital through posthospital care. Optimization of these systems can lead to better outcomes for patients after trauma. This study represents a critical first step toward better understanding the state of musculoskeletal trauma care in countries with different levels of resources, developing strategies to address deficiencies, and forming regional and international collaborations to develop musculoskeletal trauma care guidelines