Generic transversality of travelling fronts, standing fronts, and standing pulses for parabolic gradient systems

TL;DR

This paper proves that for generic potentials, traveling and standing fronts in parabolic gradient systems exhibit transversality, stability, and discrete profiles, with these properties being robust under small perturbations.

Contribution

It establishes generic transversality and stability properties of fronts and pulses in parabolic gradient systems, including robustness and discreteness of their profiles.

Findings

Traveling fronts are bistable and invade minima of V.

Standing pulses are stable at infinity.

Profiles approach limits tangentially to eigenspaces.

Abstract

For nonlinear parabolic systems of the form \[ \partial_t w(x,t) = \partial_{x}^2 w(x,t) - \nabla V\bigl(w(x,t)\bigr) \,, \] the following conclusions are proved to hold generically with respect to the potential $V$: every travelling front invading a minimum point of $V$ is bistable, there is no standing front, every standing pulse is stable at infinity, the profiles of these fronts and pulses approach their limits at $\pm\infty$ tangentially to the eigenspaces corresponding to the smallest eigenvalues of $D^2V$ at these points, these fronts and pulses are robust with respect to small perturbations of the potential, and the set of their profiles is discrete. These conclusions are obtained as consequences of generic transversality results for heteroclinic and homoclinic solutions of the differential systems governing the profiles of such fronts and pulses. Among these results, it is proved that, for a generic Hamiltonian system of the form \[ \ddot u=\nabla V(u) \,, \] every asymmetric homoclinic orbit is transverse and every symmetric homoclinic orbit is elementary.