Traveling periodic waves of chemical activity

It is shown that within the manifold of exact solutions a system of reaction-diffusion equations admits only travelling waves with planar symmetry. A derivation of the generic form of approximate (asymptotic) cylindrical and spiral travelling periodic wave solutions is given. If an exact solution homogeneous in space and periodic in time is admitted by the system of reaction-diffusion equations, then travelling periodic spiral waves are admissble as approximate solutions. This is the theoretical explanation for the travelling periodic waves of chemical activity observed in recent experiments.     

Bifurcations and new exact travelling wave solutions for the bidirectional wave equations

By using the method of dynamical system, the bidirectional wave equations are considered. Based on this method, all kinds of phase portraits of the reduced travelling wave system in the parametric space are given. All possible bounded travelling wave solutions such as dark soliton solutions, bright soliton solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, numerical simulations are conducted for dark soliton solutions, bright soliton solutions and periodic travelling wave solutions to the bidirectional wave equations. The results presented in this paper improve the related previous studies.     

Application of the (G^{\prime}/G)-expansion method for the Burgers, Burgers–Huxley and modified Burgers–KdV equations

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.     

The existence and stability of travelling waves with transition layers for some singular cross-diffusion systems

This paper is concerned with the existence and the stability of travelling waves for a class of quasilinear cross-diffusion systems describing two competing or predator-prey species. Firstly, by geometric singular perturbation method, the existence of travelling waves with transition layers is proved, which extends the results of Hosono and Mimura [Y. Hosono, M. Mimura, Singular perturbation approach to travelling waves in competing and diffusion species models, J. Math. Kyoto Univ. 22 (1982) 435–461] and Gardner [R.A. Gardner, Topological Methods Arising in the Study of Travelling Waves, Reaction–Diffusion Equations, Clarendon Press, Oxford, 1990, pp. 173–198] for non-cross-diffusion systems and Wu [Y. Wu, The existence of travelling waves for a cross-diffusion system with small parameter, Beijing Math. 3 (1997) 74–85] to more general cross-diffusion systems. Applying the stability index method introduced in Alexsander et al. [J. Alexsander, R.A. Gardner, C.K.R.T. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Die Reine Angewandte Math. 410 (1990) 167–212] and Gardner and Jones [R.A. Gardner, C.R.K.T. Jones, Stability of travelling wave solutions of diffusive predator–prey systems, Trans. Am. Soc. 327 (1991) 465–524] to the more general eigenvalue problem induced by the quasilinear cross-diffusion systems, by detailed spectral and topological analysis, the travelling waves with transition layers for the cross-diffusion systems are proved to be stable, which also extends the results of Gardner and Jones [R.A. Gardner, C.R.K.T. Jones, Stability of travelling wave solutions of diffusive predator–prey systems, Trans. Am. Soc. 327 (1991) 465–524] to the more general systems.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

Improved multiparty quantum key agreement in travelling mode

The need to simultaneously balance security and fairness in quantum key agreement (QKA) makes it challenging to design a flawless QKA protocol, especially a multiparty quantum key agreement (MQKA) protocol. When designing an MQKA protocol, two modes can be used to transmit the quantum information carriers: travelling mode and distributed mode. MQKA protocols usually have a higher qubit efficiency in travelling mode than in distributed mode. Thus, several travelling mode MQKA protocols have been proposed. However, almost all of these are vulnerable to collusion attacks from internal betrayers. This paper proposes an improved MQKA protocol that operates in travelling mode with Einstein-Podolsky-Rosen pairs. More importantly, we present a new travelling mode MQKA protocol that uses single photons, which is more feasible than previous methods under current technologies.     

Rarefaction Pulses for the Nonlinear Schrödinger Equation in the Transonic Limit

We investigate the properties of finite energy travelling waves to the nonlinear Schrödinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev–Petviashvili equation. Our results generalize an earlier result of Béthuel et al. for the two-dimensional Gross–Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of vortexless travelling waves with high energy and momentum in dimension three.     

A generic travelling wave solution in dissipative laser cavity

A large family of cosh-Gaussian travelling wave solution of a complex Ginzburg–Landau equation (CGLE), that describes dissipative semiconductor laser cavity is derived. Using perturbation method, the stability region is identified. Bifurcation analysis is done by smoothly varying the cavity loss coefficient to provide insight of the system dynamics. He’s variational method is adopted to obtain the standard sech-type and the not-so-explored but promising cosh-Gaussian type, travelling wave solutions. For a given set of system parameters, only one sech solution is obtained, whereas several distinct solution points are derived for cosh-Gaussian case. These solutions yield a wide variety of travelling wave profiles, namely Gaussian, near-sech, flat-top and a cosh-Gaussian with variable central dip. A split-step Fourier method and pseudospectral method have been used for direct numerical solution of the CGLE and travelling wave profiles identical to the analytical profiles have been obtained. We also identified the parametric zone that promises an extremely large family of cosh-Gaussian travelling wave solutions with tunable shape. This suggests that the cosh-Gaussian profile is quite generic and would be helpful for further theoretical as well as experimental investigation on pattern formation, pulse dynamics and localization in semiconductor laser cavity.     

Existence of perturbed solitary wave solutions to a model equation for water waves

We prove the existence of travelling wave solutions to a fifth order partial differential equation, which is a formal asymptotic approximation for water waves with surface tension. These travelling waves are arbitrarily small perturbations of solitary waves, but are not solitary waves themselves, because they approach small amplitude oscillations for large values of the independent variable. This result suggests that for Bond numbers less than one third, there are branches of travelling wave solutions to the water wave equations, which are perturbations of supercritical elevation solitary waves, and which bifurcate from Froude number one and Bond number one third.     

利用激光超声技术研究表面微裂纹缺陷材料的低通滤波效应

利用激光激发声表面波的理论模型,研究了被激发宽带声表面波在具有表面微裂纹缺陷金属材料上的传播特性.对具有不同形状的表面缺陷模型进行了数值分析.结果表明:表面微裂纹缺陷有明显的低通效应,缺陷深度越大高频截止频率就越低,缺陷深度与低通滤波的截止频率呈近似线性关系;缺陷的宽度增大对表面波透射能量有明显的衰减作用. 关键词： 激光超声 声表面波 有限元方法 低通滤波器     

An ignition-temperature model with two free interfaces in premixed flames†

In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numerical analysis. We prove the existence of a critical Lewis number such that the travelling wave solution is stable for values of Lewis number below the critical one and is unstable for Lewis numbers that exceed this critical value. Finally, we discuss the results of numerical simulations of a fully nonlinear system that describes the perturbation dynamics of planar fronts. These simulations reveal, in particular, some very interesting ‘two-cell’ steady patterns of curved combustion fronts.     

Travelling Wave Solutions to a Special Type of Nonlinear Evolution Equation

A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of "rank". The key idea of this method is to make use of the arbitrariness of the manifold in Painleve analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.     

Tactically-driven nonmonotone travelling waves

Models of cell invasion incorporating directed cell movement up a gradient of an external substance and carrying capacity-limited proliferation give rise to travelling wave solutions. Travelling wave profiles with various shapes, including smooth monotonically decreasing, shock-fronted monotonically decreasing and shock-fronted nonmonotone shapes, have been reported previously in the literature. The existence of tactically-driven shock-fronted nonmonotone travelling wave solutions is analysed for the first time. We develop a necessary condition for nonmonotone shock-fronted solutions. This condition shows that some of the previously reported shock-fronted nonmonotone solutions are genuine while others are a consequence of numerical error. Our results demonstrate that, for certain conditions, travelling wave solutions can be either smooth and monotone, smooth and nonmonotone or discontinuous and nonmonotone. These different shapes correspond to different invasion speeds. A necessary and sufficient condition for the travelling wave with minimum wave speed to be nonmonotone is presented. Several common forms of the tactic sensitivity function have the potential to satisfy the newly developed condition for nonmonotone shock-fronted solutions developed in this work.     

Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures

The Eckhaus stability boundaries of travelling periodic roll patterns arising in binary fluid convection is analysed using high-resolution numerical methods. We present results corresponding to three different values of the separation ratio used in experiments. Our results show that the subcritical branches of travelling waves bifurcating at the onset of convection suffer sideband instabilities that are restabilised further away in the branch. If this restabilisation is produced after the turning point of the travelling-wave branch, these waves do not become stable in a saddle node bifurcation as would have been the case in a smaller domain. In the regions of instability of the uniform travelling waves we expect to find either transitions between states of different wave number or modulated travelling waves arising in these bifurcations.     

Novel travelling wave structures: few-cycle-pulse solitons and soliton molecules

We discuss a fifth order KdV(FOKdV)equation via a novel travelling wave method by introducing a background term.Results show that the background term plays an essential role in finding new abundant travelling wave structures,such as the soliton induced by negative background,the periodic travelling wave excited by the positive background,the few-cycle-pulse(FCP)solitons with and without background,the soliton molecules excited by the background.The FCP solitons are first obuained for the FOKdV equation.     

Condensed phase combustion travelling waves with sequential exothermic or endothermic reactions

The one-dimensional propagation of a combustion wave through a premixed solid fuel for two-stage kinetics is studied. We re-examine the analysis of a single reaction travelling-wave and extend it to the case of two-stage reactions. We derive an expression for the travelling wave speed in the limit of large activation energy for both reactions. The analysis shows that when both reactions are exothermic, the wave structure is similar to the single reaction case. However, when the second reaction is endothermic, the wave structure can be significantly different from single reaction case. In particular, as might be expected, a travelling wave does not necessarily exist in this case. We establish conditions in the limiting large activation energy limit for the non-existence, and for monotonicity of the temperature profile in the travelling wave.     

Exact travelling wave solutions of a class of nonlinear diffusion equations by reduction to a quadrature

A method of constructing travelling wave solutions for nonlinear diffusion equations with polynomial nonlinearities is demonstrated. By certain assumptions some bounded travelling wave solutions are found. One of the results is the first known exact solution for the Fisher equation.     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

Travelling Solitary Wave Solutions for Generalized Time-delayed Burgers-Fisher Equation

In this paper, travelling wave solutions for the generalized time-delayed Burgers-Fisher equation are studied. By using the first-integral method, which is based on the ring theory of commutative algebra, we obtain a class of travelling solitary wavesolutions for the generalized time-delayed Burgers-Fisher equation.A minor error in the previous article is clarified.     

黏性水波振荡型行波解的存在性

利用追赶法技巧,研究了倾度为常数的宽矩形通道中黏性水波振荡型行波解的存在性,证明了当黏性系数、摩擦系数、倾度和高度比满足一定关系时,存在唯一的振荡型行波解.由于解析求解的困难,通过数值计算,给出了部分结果,最后进一步讨论了可能存在的振荡型行波解的结构. 关键词： 追赶法 黏性水波 振荡型行波解