Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation

Solutions of the Degasperis-Procesi nonlinear wave equation may develop discontinuities in finite time. As shown by Coclite and Karlsen, there is a uniquely determined entropy weak solution which provides a natural continuation of the solution past such a point. Here we study this phenomenon in detail for solutions involving interacting peakons and antipeakons. We show that a jump discontinuity forms when a peakon collides with an antipeakon, and that the entropy weak solution in this case is described by a "shockpeakon" ansatz reducing the PDE to a system of ODEs for positions, momenta, and shock strengths.     

磁流体力学方程组的激波生成

研究磁流体横向流动的一维模型,在解的强间断出现后流场的性质。利用迭代法具体构造了该方程组的强间断—激波以及问题的熵解。同时,利用激波的性质,给出了各物理参量在爆破点附近的奇性估计。     

Simulating Argon,Helium, and Nitrogen Shock Waves

We simulate the structure of the shock wave front in the range of Mach numbers from 1.5 to 10 using quasigasdynamic (QGD) equations and Navier - Stokes (NS) equations. The numerical results are compared with published experimental data. QGD and NS calculations produce close results that tightly fit the experimental data, especially for the monatomic argon and helium. The QGD algorithm is found to be more stable than the NS algorithm. __________ Translated from Prikladnaya Matematika i Informatika, No. 18, pp. 66–82, 2004.     

Shock Waves in Anisotropic Cylinders

We study small-amplitude longitudinal and torsional shock waves in circular cylinders consisting of an anisotropic medium such that the velocities of the longitudinal and torsional waves are close to each other. Previously, simple waves were considered in the same situation and conditions were found for these waves to overturn and for the corresponding shock waves to form. Here we present the study of shock waves: the shock adiabat and the evolutionary conditions. The results obtained can also be related to shock waves in unbounded media with quadratic nonlinearity.     

Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity

We consider nonlinear finite-amplitude progressive shear-flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal-density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ ? 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical-layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.     

Shock Waves in Dispersive Eulerian Fluids

The long-time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third-order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (convexity or concavity) and a general pressure law. Two mechanisms leading to the breakdown of this simple wave DSW theory for sufficiently large jumps are identified: a change in the sign of dispersion, leading to gradient catastrophe in the modulation equations, and the loss of genuine nonlinearity in the modulation equations. Large amplitude DSWs are constructed for several particular dispersive fluids with differing pressure laws modeled by the generalized nonlinear Schrödinger equation. These include superfluids (Bose–Einstein condensates and ultracold fermions) and “optical fluids.” Estimates of breaking times for smooth initial data and the long-time behavior of the shock tube problem are presented. Detailed numerical simulations compare favorably with the asymptotic results in the weak to moderate amplitude regimes. Deviations in the large amplitude regime are identified with breakdown of the simple wave DSW theory.     

Shock Waves in Plane Symmetric Spacetimes

We consider Einstein's equations coupled to the Euler equations in plane symmetry, with compact spatial slices and constant mean curvature time. We show that for a wide variety of equations of state and a large class of initial data, classical solutions break down in finite time. The key mathematical result is a new theorem on the breakdown of solutions of systems of balance laws. We also show that an extension of the solution is possible if the spatial derivatives of the energy density and the velocity are bounded, indicating that the breakdown is really due to the formation of shock waves.     

血液动力学中血管流激波与驻波的相互作用

血液动力学问题是生物力学心血管系统中的重要研究课题.血管内斑块处,血管截面和血管壁的材质发生变化,对血液流动产生重要影响.血液流动中基本波及其相互作用对探究血液流动的规律、生理学意义及与疾病的关系有着重要的意义.本文研究血液动力学血液流动简化数学模型的基本波的相互作用.血管流模型是3×3非严格双曲型方程组.构造性地得到了初值为三段常状态时,血管流问题的解,即解决了激波与驻波的相互作用问题.特别地,给出四种后前激波与驻波的相互作用的结果.     

Inhomogeneous Plane Waves in Monoclinic Crystals subject to a Bias

Here we investigate the conditions of inhomogeneous plane waves propagation in monoclinic crystals subject to initial electromechanical fields. We obtain the components of the electroacoustic tensor for the class 2, resp. m, of the monoclinic system. For particular isotropic directional bivectors we derive the decomposition of the propagation condition, and we show that the specific coefficients are similar to the case of guided wave propagation in monoclinic crystals subject to a bias. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometric Theory of Flexible and Expandable Tubes Conveying Fluid: Equations,Solutions and Shock Waves

Journal of Nonlinear Science - We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube,...     

研究金属中激波构造与衰减的一个物理模型

本文给出了研究金属中激波构造与衰减的一个物理模型.为了建立高速形变下材料的本构方程和研究激波过渡带的构造,需要考虑二个独立的理论方面.首先,将比内能分解成弹性压缩能和弹性形变能,而将形变能作为弹性应变和熵的函数展开到三阶项,其中考虑了热与机械能的耦合效应.其次,从位错动力学角度建议了一个塑性松弛函数以便描述高温、高压下塑性流动的特性.另外,本文给出了一个常微分方程组用以计算定态激波过渡带中各状态变量的分布以及激波的厚度.倘若假定在激波上熵的跳跃可以忽略,并用Hugoniot压缩模量代替等熵压缩摸量,可以获得一个分析解.最后,本文还提出了求解平板对称碰撞中激波波头衰减的一个近似方法。     

Rossby Elevation Waves in the Presence of a Critical Layer

In a previous paper, we investigated the solitary-wave-like development of small-amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean-flow velocity at a certain latitude in the β-plane. We presented a general theory for the nonlinear critical-layer theory, and illustrated it by explicitly describing the motion of a depression solitary wave (D-wave). Here, we report a continuation of that study and consider the more complex case of an elevation solitary wave (E-wave). The method involves matched asymptotic expansions between the outer flow away from the critical layer and the inner flow inside the latter, both these flows having different scalings. We showed previously that the critical-layer flow expansion diverged in the case of the E-wave on the separatrices bounding the open and closed streamlines, which led us to defer a detailed E-wave study. Thus, in this paper, we examine the motion in the additional layer located along the separatrices where this singularity is removed by using a third scaling and find that the previous undesirable distortions are discarded. The evolution equation is derived and is a Korteveg-de-Vries type-equation modified by new nonlinear terms generated by the nonlinear interactions occuring in the critical layer. This equation supports a family of E-waves provided that the mean flow obeys certain conditions. The energy exchange that occurs between the mean flow and the D or E-wave during the critical-layer formation is evaluated in the quasi-steady régime assumption.     

Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

气体动力学压差方程一类波的相互作用

该文利用一个分析不等式,得到了气体动力学压差方程前向激波追赶前向激波相互作用的结果为后向激波与前向激波相离     

三维弹性固体中冲击波传输方程的Lagrange描述

在Lagrange坐标中导出了三维非线性弹性固体中冲击波幅度在任意传播方向上的传输方程.导出的方程说明,冲击波的幅度在任意传播方向上随时间的变化率依赖于(i)冲击波阵面紧后方介质运动的一个未知量;(ii)冲击波阵面的两个主曲率;(iii)冲击波法向波速在阵面内的表面梯度;(iv)和冲击波前方介质运动有关的非齐次项,当前方介质处于均匀运动状态时此项为零.文中指出了适当选择传播矢量以简化传输方程的几种方法.我们还得到了一组与介质本构方程无关的、联系冲击波各跳跃量变化率的普适关系.     

The Stability of Attached Shock Waves in Magnetogasdynamic Flow Past a Wedge

The question of the physical significance of the new phenomenaindicated by Cabannes' work on magnetogasdynamic flow past awedge is considered from the point of view of the stabilityof the shock waves. Analytical and heuristic reasons are givensuggesting that downstream-facing shocks are stable if the upstreamflow is supersonic and unstable if it is subsonic, while upstreamfacing shocks are always to be considered unstable.     

一类非凸粘性平衡律方程的粘性激波解的存在性与稳定性

本文首先利用几何奇异摄动方法，证明了粘性系数充分小时一类非凸粘性平衡律方程的粘性冲击波的存在性，推广了原来在非线性项严格凸的条件下得到的结果．进而，利用谱分析和上下解方法，证明了对固定的小的粘性系数此类波是全局渐近指数稳定的，推广了反应扩散方程中经典的全局稳定性结果．