The velocity diagram for traveling waves

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 is an element of [sigma(-), sigma(+)], we show the existence of traveling waves phi of velocity c. The range sigma is an element of (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 >= c(+), while the case sigma = sigma(-) is negatively monostable with a branch of velocities c <= 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 (sic) c(sigma).