Existence and Non-Existence of Fisher-KPP Transition Fronts

We consider Fisher-KPP-type reaction–diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global-in-time solutions while creating a global-in-time bump-like solution. This is the first example of a medium in which no reaction–diffusion transition front exists. A weaker localized inhomogeneity leads to the existence of transition fronts, but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type reactions in inhomogeneous media.     

Regularity of Transition Fronts in Nonlocal Dispersal Evolution Equations

It is known that solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal equations. In the present paper, we establish some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities, which allows the applicability of various techniques for reaction–diffusion equations to nonlocal equations, and hence serves as an initial and fundamental step for further studying various important qualitative properties of transition fronts such as stability, uniqueness and asymptotic speeds. We also prove the existence of continuously differentiable and increasing interface location functions, which give a better characterization of the propagation of transition fronts and are of great technical importance.     

激波与火焰的相互作用过程

基于带化学反应的Navier Stokes方程和有关的热力学和反应动力学数据,利用改进的VLS格式,对甲烷 空气混合物中激波与火焰的相互作用进行了数值模拟。根据计算结果,讨论了激波掠过火焰时的变形、分叉和发展,以及激波作用下火焰的失稳、变形、破碎和相应的带旋涡的流场。     

Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.     

Rapid detonation initiation by sparks in a short duct: a numerical study

Rapid onset of detonation can efficiently increase the working frequency of a pulse detonation engine (PDE). In the present study, computations of detonation initiation in a duct are conducted to investigate the mechanisms of detonation initiation. The governing equations are the Euler equations and the chemical kinetic model consists of 19 elementary reactions and nine species. Different techniques of initiation have been studied for the purpose of accelerating detonation onset with a relatively weak ignition energy. It is found that detonation ignition induced by means of multiple sparks is applicable to auto-ignition for a PDE. The interaction among shock waves, flame fronts and the strip of pre-compressed fresh (unburned) mixture plays an important role in rapid onset of detonation.     

Thermodynamic models of phase transformations and failure waves

Dynamics and quasi-statics of heterogeneous systems with sharp interfaces are analyzed. We dwell on two particular problems: dynamics of two-layered liquid incompressible planets with phase interfaces and failure fronts in brittle solids. In the former, the dynamics of the interfaces is controlled by the equality or jump in the scalar chemical potential. Similarly, in the latter example, it is controlled by the asymmetric tensorial chemical potential. We made several simplifying assumptions to reduce the system of partial differential equations to the systems of ordinary differential equations. We briefly touch on still existing obstacles.     

A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE^2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.     

Mass,momentum and total excess energy transported by a weak planeN wave

A weakly nonlinear plane acoustic wave is emitted into an ideal gas of semi-infinite extent from an infinite plate by its sinusoidal motion of single period. The wave develops into anN wave in the far field, as long as the energy dissipation is negligible everywhere except for discontinuous shock fronts. The third-order effects at shock fronts are evaluated, i.e., the generation of reflected acoustic wave as a result of the interaction of shock and expansion wave and the production of entropy by the energy dissipation at shock fronts. Consideration of these effects enables one to estimate the whole mass, momentum and total excess energy (sum of the kinetic energy and excess of internal energy over an initial undisturbed value) transported by theN wave to the accuracy of third order of wave amplitude. It is shown that the mass and total excess energy transported by theN wave increase and the momentum decreases to asymptotic limits as the wave propagates. The result shows good agreement with a numerical result obtained by solving the Euler equations with a high-resolution TVD upwind scheme.     

The Symplectic Evans Matrix,¶and the Instability of Solitary Waves and Fronts

Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary-wave or front solutions. In this paper, we present a new symplectic framework for analysing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure – with two-form denoted by Ω– associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts. We introduce the concept of the symplectic Evans matrix, a matrix consisting of restricted Ω-symplectic forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter λ to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely on the abstract geometry of the equations, and results for the large |λ| behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case. The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation. By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained. The theory applies to a large class of solitary waves and fronts including waves which are bi-asymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schr?dinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics. Accepted August 7, 2000 ?Published online January 22, 2001     

In-plane perturbation of a system of two coplanar slit-cracks – I: Case of arbitrarily spaced crack fronts

In order to lay the grounds for a future study of the deformation of the fronts of coplanar cracks during their final coalescence, we consider the model problem of a system of two coplanar, parallel, identical slit-cracks loaded in mode I in some infinite body. The first, necessary task is to determine the distribution of the stress intensity factors along the crack fronts resulting from some small but otherwise arbitrary in-plane perturbation of these fronts. This is done here in the case where the distances between the various crack fronts are arbitrary and fixed.The first order expression of the local variation of the stress intensity factor is provided by a general formula of Rice (1989) in terms of some “fundamental kernel” tied to the mode I crack face weight function. In the specific case considered, this fundamental kernel reduces to six unknown functions; the problem is to determine them. This is done by using another formula of Rice (1989) which provides the variation of the fundamental kernel in a similar way. This second formula is applied to special perturbations of the crack fronts preserving the shape and relative dimensions of the cracks while modifying their absolute size and orientation. The output of this procedure consists of nonlinear integro-differential equations on the functions looked for, which are transformed into nonlinear ordinary differential equations through Fourier transform in the direction of the crack fronts, and then solved numerically.     

Traveling Wave Solutions for a Class of Predator–Prey Systems

We use a shooting method to show the existence of traveling wave fronts and to obtain an explicit expression of minimum wave speed for a class of diffusive predator?Cprey systems. The existence of traveling wave fronts indicates the existence of a transition zone from a boundary equilibrium to a co-existence steady state and the minimum wave speed measures the asymptotic speed of population spread in some sense. Our approach is a significant improvement of techniques introduced by Dunbar. The advantage of our method is that it does not need the notion of Wazewski??s set and LaSalle??s invariance principle used in Dunbar??s approach. In our approach, we convert the equations for traveling wave solutions to a system of first order equations by a ??non-traditional transformation??. With this converted new system, we are able to construct a Liapunov function, which gives an immediate implication of the boundedness and convergence of the relevant class of heteroclinic orbits. Our method provides a more efficient way to study the existence of traveling wave solutions for general predator?Cprey systems.     

求解变分型积分方程的一种新型数值方法——有限变分法

将作者提出的多虚拟裂纹扩展法(MVCE法)拓展为求解变分型积分方程问题的一种新型数值方法——有限变分法(FVM)。它的基本思想是,给定有限个(N个)局部变分模式,将所求解的未知量用适当的方法离散化,针对这N个局部变分模式列出N个方程,求解N个未知系数,从而求得未知量。单一未知变量FVM的最终方程组的系数矩阵通常是一个对称的窄带矩阵,对角元是大数,有很好的数值计算性能。用FVM求解了三维I型裂纹前缘的应力强度因子(SIF)分布。利用基于FVM的通用权函数法计算程序,可以高精度和高效率地求解表面力、体积力和温度载荷共同作用情况下三维裂纹前缘SIF的分布及其时间历程。FVM可以被推广到更广泛的领域,是一个求解变分型积分方程问题的普遍适用的新型数值方法。     

The Structure of Precipitation Fronts for Finite Relaxation Time

When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper’s simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax’s stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.     

Propagation of weak waves in elastic-plastic solids

Wave fronts admitting discontinuities only in the derivatives of the dependent variables are by convention called ‘weak’ waves. For the special case of discontinuous first-order derivatives, the fronts are customarily called ‘acceleration’ waves. If the governing equations are quasi-linear, then the weak waves are necessarily characteristic surfaces. Sometimes, these surfaces are also referred to as ‘singular surfaces’ of order r ? 1, where r stands for the order of the first discontinuous derivatives. This latter approach is adopted in this paper and applied to governing equations which form a set of first-order quasi-linear hyperbolic equations. When these equations are written in terms of singular surface coordinates, simplification occurs upon differencing equations written on the front and rear sides of the surface: a set of algebraic (‘connection’) equations is generated for the discontinuities in the normal derivatives of the dependent variables across the surface. When a similar operation is performed on the governing equations written for second-order derivatives, a set of first-order differential (‘transport’) equations is generated.     

Mach reflection of weak shock waves from a rigid wall

On the basis of experimental observations and theoretical analysis of flow structure in the neighborhood of the triple point, it is shown that one should reject the condition for equality of the angle of deflection of flows passing through the Mach front and the two other fronts and replace it with some supplementary condition. The system of consistency equations in the indicated region is closed by an equation which is obtained under the assumption of the extremality of the deflection angle of a flow passing through the incident and reflected fronts. Calculations of the pressure drops behind the shock fronts agree with experimental data in this case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 26–33, September–October, 1973.The authors thank S. A. Khristianovich for consideration of the work and advice.     

Diffuse interface model for high speed cavitating underwater systems

High speed underwater systems involve many modelling and simulation difficulties related to shocks, expansion waves and evaporation fronts. Modern propulsion systems like underwater missiles also involve extra difficulties related to non-condensable high speed gas flows. Such flows involve many continuous and discontinuous waves or fronts and the difficulty is to model and compute correctly jump conditions across them, particularly in unsteady regime and in multi-dimensions. To this end a new theory has been built that considers the various transformation fronts as ‘diffuse interfaces’. Inside these diffuse interfaces relaxation effects are solved in order to reproduce the correct jump conditions. For example, an interface separating a compressible non-condensable gas and compressible water is solved as a multiphase mixture where stiff mechanical relaxation effects are solved in order to match the jump conditions of equal pressure and equal normal velocities. When an interface separates a metastable liquid and its vapor, the situation becomes more complex as jump conditions involve pressure, velocity, temperature and entropy jumps. However, the same type of multiphase mixture can be considered in the diffuse interface and stiff velocity, pressure, temperature and Gibbs free energy relaxation are used to reproduce the dynamics of such fronts and corresponding jump conditions. A general model, based on multiphase flow theory is thus built. It involves mixture energy and mixture momentum equations together with mass and volume fraction equations for each phase or constituent. For example, in high velocity flows around underwater missiles, three phases (or constituents) have to be considered: liquid, vapor and propulsion gas products. It results in a flow model with 8 partial differential equations. The model is strictly hyperbolic and involves waves speeds that vary under the degree of metastability. When none of the phase is metastable, the non-monotonic sound speed is recovered. When phase transition occurs, the sound speed decreases and phase transition fronts become expansion waves of the equilibrium system. The model is built on the basis of asymptotic analysis of a hyperbolic total non-equilibrium multiphase flow model, in the limit of stiff mechanical relaxation. Closure relations regarding heat and mass transfer are built under the examination of entropy production. The mixture equation of state (EOS) is based on energy conservation and mechanical equilibrium of the mixture. Pure phases EOS are used in the mixture EOS instead of cubic one in order to prevent loss of hyperbolicity in the spinodal zone of the phase diagram. The corresponding model is able to deal with metastable states without using Van der Waals representation.     

A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell

We formulate the continuum field equations and constitutive equations that govern deformation, stress, and electric current flow in a Li-ion half-cell. The model considers mass transport through the system, deformation and stress in the anode and cathode, electrostatic fields, as well as the electrochemical reactions at the electrode/electrolyte interfaces. It extends existing analyses by accounting for the effects of finite strains and plastic flow in the electrodes, and by exploring in detail the role of stress in the electrochemical reactions at the electrode-electrolyte interfaces. In particular, we find that that stress directly influences the rest potential at the interface, so that a term involving stress must be added to the Nernst equation if the stress in the solid is significant. The model is used to predict the variation of stress and electric potential in a model 1-D half-cell, consisting of a thin film of Si on a rigid substrate, a fluid electrolyte layer, and a solid Li cathode. The predicted cycles of stress and potential are shown to be in good agreement with experimental observations.     

Spatial motion of a chemically active medium near a caustic

A study is made of the three-dimensional problem of determining the parameters of motion of a gaseous chemically active medium near a caustic, the envelope curve of the rays of the wave fronts in the geometrical acoustics approximation. Two limiting processes whereby perturbations propagate [1] can be distinguished, depending on the ratio of the reaction time of the chemical reaction to a macroscopic time: a quasifrozen process and a quasiequilibrium process. The problem is considered in a linear formulation in [2-6] in the absence of viscosity, thermal conductivity, and chemical reactions. Nonlinear equations are derived in [7–10] for an arbitrary nondissipative medium near a caustic. In the present paper Ryzhov's method [1] is used to derive the nonlinear equations of motion of the medium for both types of process. The pressure distributions near and on the caustic itself are found for an incident step wave. The effect of the chemical reaction on how the flow parameters are distributed in the vicinity of the caustic is ascertained. Equations are derived for an inhomogeneous initially moving fluid near a caustic. A nonlinear equation containing a highest derivative of third order is obtained in the vicinity of the caustic for the case of special media in which the limiting velocities of sound in the mixture at rest are close in value. It is shown that the solution of the corresponding linear equation is expressed in the form of a quadrature from the solution for a chemically inert medium and contains oscillations near the wave fronts.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 81–91, March–April, 1977.     

Focusing and Scattering of Plane Shock Waves at an Interface between Anisotropic Elastic Media

The paper is focused on the problem of constructing evolving fronts of quasilongitudinal and quasitransverse shock waves formed by incidence of an initial plane shock wave on a curvilinear interface between elastic transverse isotropic media with different physical properties. The parameter continuation method and the Newton algorithm are used to solve nonlinear Snell's equations. A method for calculating discontinuities of field functions is proposed. Shockwave scattering and focusing as a particular case of bifurcation of shock fronts and formation of caustics are considered. A numerical example is given.     

Shock and acceleration waves with large amplitudes in laminated composite plates

Propagation of shock and acceleration waves with large amplitudes is studied. The geometrical nonlinearity in the von Karman sense is included in deriving the plate equations. The dynamical conditions on the wave fronts are derived from the three-dimensional conditions in a way consistent with the derivation of the plate equations. General equations governing the propagation velocities are obtained. Solutions are presented for the case where the plates are initially at rest. It is found that, in this case, the large amplitude has a substantial effect only on the transverse shear shock wave. Finally, stability of the wave front is discussed.