Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

Abstract

Abstract. The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in R 2 with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds.The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.

Traveling Fronts in a Reaction–Diffusion Equation with a Memory Term

“…A significant body of work (cf. [3,6,9,18,19,23,29]) considers the propagation of waves in periodic media. If the period of the oscillating coefficient is very small one can perform a homogenization and consider traveling waves in the homogenized system.…”

“…In [23] reaction-diffusion systems are studied and exponential averaging is used to show that traveling wave solutions can be described by a spatially homogeneous equation and exponentially small remainders. The approach based on center-manifold reduction in [6] applies to traveling waves in parabolic equations and, moreover, the authors prove the existence of a generalized oscillating wave that converges to a limiting wave. We point out that all previously mentioned articles study limit problems of "one-scale" nature, in contrast to [17] and the present work where traveling pulses in "two-scale FitzHugh-Nagumo systems" are investigated, see Sect.…”

Traveling fronts in a reaction-diffusion equation with a memory term

“…In [MSU07] reaction-diffusion systems are studied and exponential averaging is used to show that traveling wave solutions can be described by a spatially homogeneous equation and exponentially small remainders. The approach based on center-manifold reduction in [BoM14] applies to traveling waves in parabolic equations and, moreover, the authors prove the existence of a generalized oscillating wave that converges to a limiting wave. We point out that all previously mentioned articles study limit problems of "one-scale" nature, in contrast to [GuR18] and the present work where traveling pulses in "two-scale FitzHugh-Nagumo systems" are investigated, see Section 3.3.…”

Pulses in FitzHugh--Nagumo Systems with Rapidly Oscillating Coefficients

“…In [MSU07] reaction-diffusion systems are studied and exponential averaging is used to show that traveling wave solutions can be described by a spatially homogeneous equation and exponentially small remainders. The existence of generalized (oscillating) traveling waves u ε (t, x) = u(x + ct, x ε ) and their convergence to a limiting wave U (t, x) = u 0 (x + ct) is proved for parabolic equations in [BoM14]. In their approach, the authors reformulate the problem as a spatial dynamical system and use a centre manifold reduction.…”