Related papers: Singular Shocks in a Chromatography Model
Our object of study is non smooth vector fields on $\R^2$. We apply the techniques of geometric singular perturbations in non smooth vector fields after regularization and a blow$-$up. In this way we are able to bring out some results that…
We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approximate model of the…
A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…
We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave…
We study a simplification of the well-known Shigesada-Kawasaki-Teramoto model, which consists of two nonlinear reaction-diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an…
We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor…
We consider the stability of a system of equations which are a singular perturbation of the incompressible rigid-plastic flow equations used to model granular flow. A linear stability analysis shows that solutions of these equations are…
A system of two-dimensional nonlinear equations of hydrodynamics is considered. It is shown that for the this system in the general case a solution with weak discontinuity-type singularity behaves as a square root of S(x,y,t), where…
Reaction-nonlinear diffusion (RND) partial differential equations are a fruitful playground to model the formation of sharp travelling fronts, a fundamental pattern in nature. In this work, we demonstrate the utility and scope of…
We illustrate an alternative derivation of the viscous regularization of the diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. {\em Trans. Amer. Math. Soc.}, 324:331--351]. We provide an alternative proof of existence…
An asymptotic method for finding instabilities of arbitrary $d$-dimensional large-amplitude patterns in a wide class of reaction-diffusion systems is presented. The complete stability analysis of 2- and 3-dimensional localized patterns is…
The incompatibilities between the initial and boundary data will cause singularities at the time-space corners, which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions. We study the corner…
The paper is aimed at analysing a singular perturbation of the Navier-Stokes equations on a compact closed manifold. The case of compact smooth manifolds with boundary under the Dirichlet conditions is also included. Global existence and…
A parameter dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non-self-adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet…
We consider reaction-diffusion equations on the planar square lattice that admit spectrally stable planar travelling wave solutions. We show that these solutions can be continued into a branch of travelling corners. As an example, we…
Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable…
It is shown how a recent method to systematically extrapolate and resum the loop expansion for nonlinear sigma-models is related to solutions of the renormalization group equation. This relation is used to generalize the explicit equations…
This chapter is mainly a tutorial introduction to methods recently developed in order to find all (as opposed to some) meromorphic particular solutions of given nonintegrable, autonomous, algebraic ordinary differential equations of any…
The work is devoted to the construction of the asymptotic behavior of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends…
Recently, a novel bifurcation technique known as the deflated continuation method (DCM) was applied to the single-component nonlinear Schr\"odinger (NLS) equation with a parabolic trap in two spatial dimensions. The bifurcation analysis…