Stability properties for solitary waves in 3 space dimensions

In this paper we study a Lorentz invariant equation in three space dimensions which has solitary wave solutions. These solutions are labelled by means of a topological invariant which allows to prove suitable stability properties.     

Stability calculation of travelling waves in a simplified brake model

We consider a system composed of an elastic tube, which is fixed at the outer boundary and in frictional contact with a rigid cylinder, rotating inside the tube about the common axis. Using a 1–mode Galerkin ansatz in radial direction, a non–smooth PDE for the tangential and radial deformations at the contact between the two bodies is obtained. It has been shown ([1]) that different types of travelling stick–slip–separation waves exist, but these waves are unstable for most parameter values. In this investigation we first enlarge the model by viscuous damping and second we introduce a larger number of nodes in radial direction for the numerical analysis. The influence of these two effects on the shape and stability of the travelling waves is studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of travelling waves in models of superconductivity

The linear stability of two travelling waves describing phasechanges in superconducting materials is examined. The firstdescribes the change of phase between normally conducting andsuperconducting states in a type I superconductor, while thesecond describes the transition from the superconducting stateto the mixed state in a type II superconductor.     

Existence of travelling waves in a modified vector-disease model

In this paper, we consider travelling wave solutions for a modified vector-disease model. Special attention is paid to the model in which a susceptible vector can receive the infection not only from the infectious host but also from the infectious vector. For the strong generic delay kernel, we show that travelling wave solutions exist using the geometric singular perturbation theory.     

Stability and travelling waves for a time-delayed population system with stage structure

We study a time-delayed population system with stage structure for the interaction between two species, the adult members of which are in competition. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. The global stability results are established for each equilibrium. The criteria for global convergence to each equilibrium are sharp and involve these delays. By using lower and upper travelling wave solutions, we show that the model has travelling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.     

Stability of periodic traveling flexural‐gravity waves in two dimensions

In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.     

Stability of travelling waves for non-convex scalar viscous conservation laws

Travelling waves of a viscous conservation law (so-called viscous profiles) are shown to be stable in polynomialy weighted L^∞ spaces. There is no assumption of convexity on the nonlinear term and thus earlier results of Il'in and Oleǐnik, Sattinger, and Kawashima and Matsumura are generalized. The method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula. Estimates are derived for the semigroup through a new technique for estimating the resolvent. © 1993 John Wiley & Sons, Inc.     

The isentropic quantum drift-diffusion model in two or three space dimensions

We investigate the isentropic quantum drift-diffusion model, a fourth order parabolic system, in space dimensions d = 2, 3. First, we establish the global weak solutions with large initial value and periodic boundary conditions. Then we show the semiclassical limit by delicate interpolation estimates and compactness argument.     

The isentropic quantum drift-diffusion model in two or three space dimensions

We investigate the isentropic quantum drift-diffusion model, a fourth order parabolic system, in space dimensions d = 2, 3. First, we establish the global weak solutions with large initial value and periodic boundary conditions. Then we show the semiclassical limit by delicate interpolation estimates and compactness argument. Supported by the Natural Science Foundation of China (No.10401019 and No.10626030).     

Stability of the sum of two solitary waves for (gDNLS) in the energy space

In this paper, we continue the study in [18]. We use the perturbation argument, modulational analysis and the energy argument in [15], [16] to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schrödinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case $\sigma >1$ we considered corresponds to the $L^2$-supercritical case.     

The Cauchy problem for a spin-liquid model in three space dimensions

In this paper, we study the Cauchy problem for magnetic fluid of spin-liquid type with Mermin-Ho relation in the three-dimensional space and prove global existence and uniqueness of solutions. The proof is based on the equivalence relation between smooth solutions of the spin-liquid model and the systems of Schrödinger equations with Abelian gauge field. The Sobolev spaces with fractional derivatives are also used in our estimates.     

Detonatlve travelling waves for combustions

The existence of travelling wave solution to equations of a viscous heat conducting combustible fluid is proved. The reactions are assumed to be one step exothermic reactions with a natural discontinuous reaction rate function. The problem is studied for a general gas. Instead of assuming the ideal gas conditions we consider a general thermodynamics which is described by a fairly mild set of hypotheses. Travelling waves for detonations reduce to specific heteroclinic orbits of a discontinuous system of ODE's. The existence proof for heteroclinic orbits corresponding to weak and strong detonation waves is carried out by some general topological arguments in ODE. The uniqueness and nonuniqueness of these waves are also considered.     

Convergence rates toward the travelling waves for a model system of the radiating gas

The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t)^?1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd.     

Forms of travelling waves admitted by a mechanochemical model of tumour angiogenesis

We study the existence and the properties of travelling wave solutions in a system of non‐linear partial differential equations, which arise in some mechanochemical models of angiogenesis and/or vasculogenesis. Under the ‘weak traction’ assumption we prove the existence and uniqueness (up to a translation) of solutions. We show the positiveness of the endothelial cell density and determine its asymptotic behaviour; also we show that the TAF concentration function is positive and monotonic. Copyright © 2010 John Wiley & Sons, Ltd.