event
Tuesday 28 Feb 2012: Existence and stability of stationary fronts in inhomogeneous wave equations
Gianne Derks - University of Surrey
St John's Room, Kay Building 16:00-17:00
Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. This talk considers the effects of (non-local) inhomogeneities on the existence and stability of fronts in nonlinear wave equations.
Homogeneous nonlinear wave equations are Hamiltonian partial differential equations with the homogeneity providing an extra symmetry in the form of the spatial translations. Inhomogeneities break the translational symmetry, though the Hamiltonian structure is still present. When the spatial translational symmetry is broken, travelling waves are no longer natural solutions. Instead, the travelling waves tend to interact with the inhomogeneity and get trapped, reflected, or slowed down. As a starting point in understanding this interaction, we study the existence and stability of stationary fronts in the inhomogeneous wave equations.
We look at wave equations with finite length inhomogeneities and assume that the spatial domain can be written as the union of disjoint intervals, such that on each interval the wave equation is homogeneous. The underlying Hamiltonian structure allows for a rich family of stationary front solutions. The existence of stationary fronts is shown by connecting the unstable respectively stable manifolds of the fixed points in the extremal intervals with the Hamiltonian orbits in the intermediate intervals. So the values of the energy (Hamiltonian) in each intermediate interval provide natural parameters for the family of orbits. After having established existence, the next question is the stability of those stationary fronts. Using Evans function type arguments, we show that changes of stability can only occur at critical points of the length of the inhomogeneity as a function of the energy density inside the inhomogeneity. We also give expressions for the related eigenfunctions in terms of the front solution. With this observation, the stability of the fronts can often be determined by using continuation arguments and Sturm-Liouville theory.
To illustrate how these results can be used to find families of stationary fronts and their stability, we consider two models involving the inhomogeneous sine-Gordon equation. One model describes aspects of the DNA/RNAP interaction in DNA copying. The other model involves long Josephson junctions or 0-$\pi$ junctions with imperfections. The existence of rich families of stationary coherent solutions will be discussed, including stable non-monotonic fronts.
