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.     

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.     

The Asymptotic Speed of Spread and Traveling Waves for a Vector Disease Model

The asymptotic speed of spread is established for a diffusive and time-delayed integro-differential equation modeling vector disease, and its coincidence with the minimal wave speed for monotone traveling waves is proved. An erratum to this article can be found at     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Axisymmetric and Other States

A rigorous approach of nonlinear continuum mechanics is used to derive nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. Nonlinearity is introduced by means of metric coefficients, the Cauchy—Green strain tensor, and the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. Quadratically nonlinear wave equations are derived for three states (configurations): (i) axisymmetric configuration dependent on the radial and axial coordinates and independent of the angular coordinate, (ii) configuration dependent on the angular coordinate, and (iii) axisymmetric configuration dependent on the radial coordinate. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. Six different systems of wave equations are written __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 72–84, June 2005.