RESUMO
We study the effect of a small cutoff epsilon on the velocity of a pulled front in one dimension by means of a variational principle. We obtain a lower bound on the speed dependent on the cutoff, for which the two leading order terms correspond to the Brunet-Derrida expression. To do so we cast a known variational principle for the speed of propagation of fronts in different variables which makes it more suitable for applications.
RESUMO
We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction-diffusion equation when a small cutoff is applied to the reaction term at the unstable or metastable equilibrium point. The results are valid for arbitrary reaction terms and include the case of density-dependent diffusion.
RESUMO
We study the minimal speed of propagating fronts of convection-reaction-diffusion equations of the form u(t)+microphi(u)u(x)=u(xx)+f(u) for positive reaction terms with f(')(0)>0. The function phi(u) is continuous and vanishes at u=0. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assessment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if f(")(u)/sqrt[f(')(0)]+microphi(')(u)<0, then the minimal speed is given by the linear value 2sqrt[f(')(0)], and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case u(t)+microuu(x)=u(xx)+u(1-u). Results are also given for the density dependent diffusion case u(t)+microphi(u)u(x)=[D(u)u(x)](x)+f(u).
RESUMO
We study traveling fronts of equations of the form u(tt)+phi(u)u(x)=u(xx)+f(u). A criterion for the transition from linear to nonlinear marginal stability is established for positive functions phi(u) and for any reaction term f(u) for which the usual parabolic reaction diffusion equation u(t)=u(xx)+f(u) admits a front. As an application, we treat reaction diffusion systems with transport memory.
RESUMO
We show that the amplitude of the limit cycle of Rayleigh's equation can be obtained from a variational principle. We use this principle to reobtain the asymptotic values for the period and amplitude of the Rayleigh and van der Pol equations. Limit cycles of general Liénard systems can also be derived from a variational principle.