Generalized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front. In the case of stationary ergodic reactions, the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.     

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay

Existence,Uniqueness and Asymptotic Stability of Traveling Wavefronts in A Non-Local Delayed Diffusion Equation

In this paper, we study the existence, uniqueness, and global asymptotic stability of traveling wave fronts in a non-local reaction–diffusion model for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. Under realistic assumptions on the birth function, we construct various pairs of super and sub solutions and utilize the comparison and squeezing technique to prove that the equation has exactly one non-decreasing traveling wavefront (up to a translation) which is monotonically increasing and globally asymptotic stable with phase shift.      

Stability of Concatenated Traveling Waves

We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is individually asymptotically stable. We show that the sequence of traveling waves is itself asymptotically stable: as \(t\rightarrow \infty \), the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Essentially the same result was previously proved by Wright (J Dyn Differ Equ 21:315–328, 2009) and Selle (Decomposition and stability of multifronts and multipulses, 2009), who regarded the concatenated wave pattern as a sum of traveling waves. In contrast to their work, we regard the pattern as a sequence of traveling waves restricted to subintervals of \(\mathbb {R}\) and separated at any finite time by small jump discontinuities. Our proof uses spatial dynamics and Laplace transform.     

Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction

We develop a singular perturbation technique to study the existence of periodic traveling wave solutions with large wave speed for a class of reaction-diffusion equations with time delay and non-local response. Unlike the classical singular perturbation method, our approach is based on a transformation of the differential equations to integral equations in a Banach space that reduces the singular perturbation problem to a regular perturbation problem. The periodic traveling wave solutions then are obtained by the use of Liapunov-Schmidt method and a generalized implicit function theorem. The general result obtained has been applied to a non-local reaction-diffusion equation derived from an age-structured population model with a logistic type of birth function.     

Existence of Traveling Waves of General Gray-Scott Models

This work gives a rigorous proof of the existence of propagating traveling waves of a nonlinear reaction–diffusion system which is a general Gray-Scott model of the pre-mixed isothermal autocatalytic chemical reaction of order m (\(m > 1\)) between two chemical species, a reactant A and an auto-catalyst B, \( A + m B \rightarrow (m+1) B\), and a super-linear decay of order \( n > 1\), \( B \rightarrow C\), where \( 1< n < m\). Here C is an inert product. Moreover, we establish that the speed set for existence must lie in a bounded interval for a given initial value \(u_0\) at \( - \infty \). The explicit bound is also derived in terms of \(u_0\) and other parameters. The same system also appears in a mathematical model of SIR type in infectious diseases.     

Traveling Waves in Porous Media Combustion: Uniqueness of Waves for Small Thermal Diffusivity

We study traveling wave solutions arising in Sivashinsky’s model of subsonic detonation which describes combustion processes in inert porous media. Subsonic (shockless) detonation waves tend to assume the form of a reaction front propagating with a well defined speed. It is known that traveling waves exist for any value of thermal diffusivity [5]. Moreover, it has been shown that, when the thermal diffusivity is neglected, the traveling wave is unique. The question of whether the wave is unique in the presence of thermal diffusivity has remained open. For the subsonic regime, the underlying physics might suggest that the effect of small thermal diffusivity is insignificant. We analytically prove the uniqueness of the wave in the presence of non-zero diffusivity through applying geometric singular perturbation theory. Dedicated to Mr. Brunovsky in honor of his 70th birthday.     

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.     

Existence,Uniqueness, and Asymptotic Stability for a Thermoelastic Plate

In this note, we are concerned with the linear theory of a thermoelastic plate when a rate-type equation is assumed for the heat flux. We consider an initial boundary-value problem for this plate and show the existence, uniqueness, and asymptotic stability of a solution. Thermodynamic restrictions on the assumed constitutive equations are also derived. Finally, we give an expression for the pseudo-free energy.     

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.     

Solitary Waves in Nonlinear Beam Equations: Stability,Fission and Fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

Time Periodic Traveling Waves for a Periodic and Diffusive SIR Epidemic Model

In this paper, we study the time periodic traveling wave solutions for a periodic SIR epidemic model with diffusion and standard incidence. We establish the existence of periodic traveling waves by investigating the fixed points of a nonlinear operator defined on an appropriate set of periodic functions. Then we prove the nonexistence of periodic traveling via the comparison arguments combined with the properties of the spreading speed of an associated subsystem.     

Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media

We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background ${(\mathbb{R}^{3 + 1}, g)}$ with a time dependent metric g coinciding with the Minkowski metric outside the cylinder ${\{(t, x) || x | \leq R\}}$ . We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work in the particular case when g is merely C ¹ close to the Minkowski metric. One of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric.     

Collapsing and Explosion Waves in Phase Transitions with Metastability,Existence, Stability and Related Riemann Problems

Collapsing waves were observed numerically before and were used to explain the ring formations in dynamic flows involving phase transitions with metastability. In this paper, necessary and sufficient conditions for collapsing type of waves to exist are given. The conditions are that the wave speed of the collapsing wave is not less than a number and is supersonic on both sides of the wave. Existence and non-existence conditions for the explosion waves are also found. The stability of these waves are studied numerically. Although there are infinitely many collapsing (or explosion) waves for a fixed downstream state, the collapsing (or explosion) wave appeared in the solution of Riemann problem is numerically verified to be the one with the slowest speed. Although a Riemann problem in the zero viscosity limit may have two solutions, one with, the other without, a collapsing (or explosion) wave, from the vanishing viscosity point of view, the one with a collapsing (or explosion) wave is numerically verified to be admissible.     

Stability of Solitary Waves for the Generalized Higher-Order Boussinesq Equation

This work studies the stability of solitary waves of a class of sixth-order Boussinesq equations.     

Existence and Conditional Energetic Stability of Capillary-Gravity Solitary Water Waves by Minimisation

Firstly, the two-dimensional stationary water-wave problem is considered. Existence of capillary-gravity solitary waves is proved by minimising a functional related to Smales amended potential. We first establish the existence of periodic solutions of arbitrarily large periods, leading to a minimising sequence in L²() that stays away from the boundary of the neighbourhood of 0 W^2,2() in which the analysis is carried out. With the help of the concentration-compactness principle, we then show that every minimising sequence has a subsequence that, after possible shifts in the propagation direction, converges in L²() to a minimiser. Secondly, for the evolutionary problem, we prove that the set of minimal solitary waves as a whole is energetically conditionally stable. Energetically means that the distance to the set of all minimisers is defined in terms of the total energy, and conditionally means that we consider solutions to the evolutionary problem that do not explode instantaneously but could perhaps explode in finite time (e.g., via the explosion of another norm). We work in some bounded set in W^2,2() that contains the quiescent state and we are not interested in the fate of solutions that leave this set.     

A Variant of Newton's Method for the Computation of Traveling Waves of Bistable Differential-Difference Equations

We consider a variant of Newton's method for solving nonlinear differential-difference equations arising from the traveling wave equations of a large class of nonlinear evolution equations. Building on the Fredholm theory recently developed by Mallet-Paret we prove convergence of the method. The utility of the method is demonstrated with a series of examples.     

Large Time Behavior and Stability of Equilibria of Degenerate Parabolic Equations

The article deals with positive solutions of the Dirichlet problem for

Periodic Solutions of Delay Differential Equations with a Small Parameter: Existence,Stability and Asymptotic Expansion

We consider a scalar delay differential equation with a small parameter, and employ Walthers method to obtain a result on the existence and stability of a slowly oscillatory periodic solution that represents a refinement of the estimate for the Lipschitz constant of a returning map. We also develop a matching method and obtain asymptotic expansions of the slowly oscillatory periodic solution and its minimal period.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthdayAMS subject classifications: 34K15; 34K20; 34C25.