Existence and stability traveling wave solutions for a system of social outbursts

Abstract

We study the existence and stability of planar traveling wave solutions to the system of reaction-diffusion equations:{ut=∂2u∂x2+Γu(1−u)1+e−β(v−1)−u,x∈Rn,t>0,vt=∂2v∂x2−v(1+u)p+1,x∈Rn,t>0, which was introduced in [8] as a model for the spatiotemporal dynamics of rioting activity and social tension. The “wave-like” spread of rioting activity observed in the 2005 French Riots [12] motivates the study of these type of solutions, which were numerically analyzed in [46]. Of particular interest is the qualitative difference in traveling wave solutions between the p<0 and p≥0 case. We present results for the existence and stability of these type of solutions for p≥0 and provide the asymptotic approach rates of the traveling wave solutions to the steady states as |x|→∞. For the case of p<0 we study the stability of such solutions using Evans Functions and a combination of analytical and numerical methods to show irrefutable evidence of spectral stability.