We consider general reaction dusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbations. We employ recently developed techniques involving point-wise Green's functions estimates for functional erential equations of mixed type (MFDEs), allowing our results to be applied even in situations where comparison principles are not available.
Autonomously gathering aerial data has many potential applications, from co-scouting with ground robots to providing live field information to dismounted units. The ARL team developed a number of autonomous multirotor capabilities, creating an infrastructure for further work at ARL and at Olin.
We study the interaction of small amplitude, long-wavelength solitary waves in the Fermi–Pasta–Ulam model with general nearest-neighbour interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intermediate values of time these solutions describe the interaction of two counter-propagating pulses. These solutions are stable with respect to perturbations in ℓ2 and asymptotically stable with respect to perturbations which decay exponentially at spatial ±∞.