Travelling Waves for Fourth Order Parabolic Equations

Abstract

Abstract. We study travelling wave solutions for a class of fourth order parabolic equations. Travelling wave fronts of the form u(x, t) = U (x + ct), connecting homogeneous states, are proven to exist in various cases: connections between two stable states, as well as connections between an unstable and a stable state, are considered.

Existence and Stability of Traveling Fronts in the Extended Fisher–Kolmogorov Equation

“…Our work extends that of [3] in several ways. First of all, we consider a general class of nonlinear terms and we develop a geometrical proof of the existence of these fronts as opposed to the variational approach in [3]. More importantly, we also analyse the nonlinear stability of these fronts which does not yet appear to have been discussed.…”

“…Also, periodic solutions , solitons (homoclinic/heteroclinic orbits) and chaotic patterns have been studied for the stationary part of this equation, see [20,26]. For the nonlinearity N=a(1 − B 2 ) − B 3 (0 < a < 1) the existence of traveling waves connecting B= ± 1 has been studied in [1,3]. Finally, in [3] it is also shown that for N(B)=−B 2 there exists a traveling wave from B=1 to B=0 for every positive wavespeed and D ¥ R.…”

“…For the nonlinearity N=a(1 − B 2 ) − B 3 (0 < a < 1) the existence of traveling waves connecting B= ± 1 has been studied in [1,3]. Finally, in [3] it is also shown that for N(B)=−B 2 there exists a traveling wave from B=1 to B=0 for every positive wavespeed and D ¥ R.…”

“…For questions of pattern formation unstable-stable fronts seem to be most relevant and we will focus exclusively on them in this article. The prior studies of fronts in the eFKequation [1,3,25,27] focussed on the existence of fronts connecting B=+1 to B=−1 which are stable-stable fronts. As far as we know the only other rigorous work on unstable-stable fronts in the eFK-equation is [3], which proves that such fronts exist if one chooses a nonlinear term N(B)=−B 2 and which proves the existence of fronts connecting B=−a to B= ± 1 for N(B)=a(1 − B…”

Homoclinic and Heteroclinic Bifurcations in Vector Fields

“…Such orbits have also been found in four-dimensional systems where the primary homoclinic orbit converges to a center with non-semisimple eigenvalues on the imaginary axis [43] and in systems that have two primary homoclinic orbits to the same saddle equilibrium [39]. In [378,407,409], variational methods were used to find multibump orbits in the Swift-Hohenberg equation.…”