The velocity diagram for traveling waves - Al Haj, Mohammad; Monneau, Régis (Comptes Rendus. Mathématique)
In this Note, we consider traveling waves in a reaction-diffusion equation in dimension one. Motivated by the motion of dislocations in crystals, we introduce an additive parameter $\sigma $ in the reaction term, which may be interpreted as an exterior force applied on the crystal. Under certain natural assumptions and for every value of $\sigma \in [\sigma ^-,\sigma ^+]$, we show the existence of traveling waves $\phi $ of velocity $c$. The range $\sigma \in (\sigma ^-,\sigma ^+)$ corresponds to bistable cases with a unique velocity $c=c(\sigma )$. On the contrary, the case $\sigma =\sigma ^+$ is positively monostable with a branch of velocities $c\ge c^+$, while the case $\sigma =\sigma ^-$ is negatively monostable with a branch of velocities $c\le c^-$. This study gives rise to a natural connection between bistable cases and monostable cases in a single velocity diagram. We also give some qualitative properties of the velocity function $\sigma \mapsto c(\sigma )$.