Folds, canards and shocks in advection-reaction-diffusion models.

*(English)*Zbl 1213.34073The authors study the existence of traveling wave solutions in advection-reaction-diffusion systems by using geometric singular perturbation techniques. Traveling waves with smooth and sharp interfaces are considered, both are well known features in tactically driven cell movement modeled by coupled advection-reaction-diffusion equations.

The authors present general conditions for the properties of ARD models under which traveling waves with smooth or sharp interfaces can be observed. In particular they show that a traveling wave analysis under the appropriate Liénard transformation reveals a generic fold condition to observe shock like interfaces in the wave form corresponding to canard solutions in the traveling wave equation. In addition, these folds in the slow equation that allow a heteroclinic connection to pass transversal to the fold are identified as ‘walls of singularities’ and ‘holes in the wall’ in the framework of tactically driven cell migration.

Their geometrical approach automatically recovers entropy conditions such as the Rankine-Hugoniot and Lax conditions for shocks in hyperbolic PDE theory.

The authors present general conditions for the properties of ARD models under which traveling waves with smooth or sharp interfaces can be observed. In particular they show that a traveling wave analysis under the appropriate Liénard transformation reveals a generic fold condition to observe shock like interfaces in the wave form corresponding to canard solutions in the traveling wave equation. In addition, these folds in the slow equation that allow a heteroclinic connection to pass transversal to the fold are identified as ‘walls of singularities’ and ‘holes in the wall’ in the framework of tactically driven cell migration.

Their geometrical approach automatically recovers entropy conditions such as the Rankine-Hugoniot and Lax conditions for shocks in hyperbolic PDE theory.

Reviewer: Julia Ehrt (Berlin)

##### MSC:

34E15 | Singular perturbations for ordinary differential equations |

34C40 | Ordinary differential equations and systems on manifolds |

35L67 | Shocks and singularities for hyperbolic equations |

92C17 | Cell movement (chemotaxis, etc.) |

34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |

35C07 | Traveling wave solutions |