The stability of traveling waves with non-critical speeds for some competition systems

This paper is concerned with some quasilinear cross-diffusion systems which model competing species in mathematical ecology.By detailed spectral analysis,each traveling wave solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted spaces.     

一类三种群捕食者-食饵交错扩散模型整体解的存在性

应用能量估计方法和Gagliardo-Nirenberg型不等式证明含一类食饵种群和两类竞争捕食者种群的反应扩散模型整体解的存在性和一致有界性,该模型是带自扩散和交错扩散项的三种群捕食者-食饵模型.     

Stable periodic traveling waves for a predator-prey model with non-constant death rate and delay

In this paper we will consider a predator-prey model with a non-constant death rate and distributed delay, described by a partial integro-differential system. The main goal of this work is to prove that the partial integro-differential system has periodic orbitally asymptotically stable solutions in the form of periodic traveling waves; i.e. N(x, t) = N(σt − μ · x), P(x, t) = P(σt − μ · x), where σ > 0 is the angular frequency and μ is the vector number of the plane wave, which propagates in the direction of the vector μ with speed c = σ/∥μ∥; and N(x, t) and P(x, t) are the spatial population densities of the prey and the predator species, respectively. In order to achieve our goal we will use singular perturbation’s techniques.     

Existence and stability of traveling waves for an integro-differential equation for slow erosion

We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order nonlinear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at ±∞. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approach the traveling waves asymptotically as t→+∞$t\to +\infty$.     

Bounded traveling waves of the oceanic currents motion equations

The bifurcation methods of differential equations are employed to investigate traveling waves of the oceanic currents motion equations. The sufficient conditions to guarantee the existence of different kinds of bounded traveling wave solutions are rigorously determined. Further, due to the existence of a singular line in the corresponding traveling wave system, the smooth periodic traveling wave solutions gradually lose their smoothness and evolve to periodic cusp waves. The results of numerical simulation accord with theoretical analysis.     

Stability of traveling waves for Hamilton-Jacobi equations with finite speed perturbations

We prove nonlinear stability of planar shock for general Hamilton-Jacobi equations with finite speed perturbation. Here we use energy estimates. It is shown that the solution connecting a weak shock is asymptotically stable under small perturbations.     

Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces.     

三级营养食物链交错扩散模型的整体解

应用能量估计方法和Gagliardo-Nirenberg型不等式证明了带自扩散和交错扩散项的三级营养食物链模型在齐次Neumann边值条件下整体解的存在唯一性和一致有界性.     

Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves

Continuing a line of investigation initiated in [F. Gesztesy, Y. Latushkin, K.A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal. 186 (2007) 361–421] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman–Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which appears to offer applications in the multi-dimensional case.A second main result is to show that the multi-dimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [G.J. Lord, D. Peterhof, B. Sandstede, A. Scheel, Numerical computation of solitary waves in infinite cylindrical domains, SIAM J. Numer. Anal. 37 (2000) 1420–1454] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [F. Gesztesy, Y. Latushkin, K.A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal. 186 (2007) 361–421] and answering a question of [J. Niesen, Evans function calculations for a two-dimensional system, presented talk, SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, USA, May 2007] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.     

Multidimensional stability of traveling fronts in monostable reaction-difusion equations with complex perturbations

This paper studies the multidimensional stability of traveling fronts in monostable reaction-difusion equations,including Ginzburg-Landau equations and Fisher-KPP equations.Eckmann and Wayne(1994)showed a one-dimensional stability result of traveling fronts with speeds c c(the critical speed)under complex perturbations.In the present work,we prove that these traveling fronts are also asymptotically stable subject to complex perturbations in multiple space dimensions(n=2,3),employing weighted energy methods.     

Asymptotic stability of traveling waves in a discrete convolution model for phase transitions

In a recent paper [P. Bates, A. Chmaj, A discrete convolution model for phase transition, Arch. Rational Mech. Anal. 150 (1999) 281-305], a discrete convolution model for Ising-like phase transition has been derived, and the existence, uniqueness of traveling waves and stability of stationary solution have been studied. This nonlocal model describes l₂-gradient flow for a Helmholts free energy functional with general range interaction. In this paper, by using the comparison principle and the squeezing technique, we prove that the traveling wavefronts with nonzero speed is globally asymptotic stable with phase shift.     

On the stability of solitary-wave solutions of model equations for long waves

Summary After a review of the existing state of affairs, an improvement is made in the stability theory for solitary-wave solutions of evolution equations of Korteweg-de Vries-type modelling the propagation of small-amplitude long waves. It is shown that the bulk of the solution emerging from initial data that is a small perturbation of an exact solitary wave travels at a speed close to that of the unperturbed solitary wave. This not unexpected result lends credibility to the presumption that the solution emanating from a perturbed solitary wave consists mainly of a nearby solitary wave. The result makes use of the existing stability theory together with certain small refinements, coupled with a new expression for the speed of propagation of the disturbance. The idea behind our result is also shown to be effective in the context of one-dimensional regularized long-wave equations and multidimensional nonlinear Schr?dinger equations.     

Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations

This work is concerned with stability properties of periodic traveling waves solutions of the focusing Schrödinger equation

iut+uxx+²|u|u=0     

On existence and stability of solutions for higher order semilinear Dirichlet problems

We provide existence and stability results for semilinear Dirichlet problems with nonlinearity satisfying general growth conditions. We consider the case when both the coefficients of the differential operator and the nonlinear term depend on the numerical parameter. We show applications for the fourth order semilinear Dirichlet problem.     

Existence,orbital stability and instability of solitary waves for coupled BBM equations

This paper is concerned with the orbital stability/instability of solitary waves for coupled BBM equations which have Hamiltonian form. The explicit solitary wave solutions will be worked out first. Then by detailed spectral analysis and decaying estimates of solutions for the initial value problem, we obtain the orbital stability/instability of solitary waves.     

Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

Orbital stability of standing waves for semilinear wave equations with indefinite energy

The orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole Rn with n?2. The stability is determined by the convexity on ω of the lowest energy d(ω) of standing waves with frequency ω. The arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows.