Elementary proof of wavetrain families near homogeneous oscillations without singular perturbation theory

Develop an elementary proof, avoiding geometric singular perturbation theory, of the existence of families of small-wavenumber wavetrains bifurcating from a spatially homogeneous periodic orbit in reaction–diffusion systems u_t = D u_{xx} + f(u). Specifically, show that near a homogeneous oscillation with simple temporal Floquet multiplier ρ = 1, there exist wavetrains parameterized by spatial wavenumber k with nonlinear dispersion relation ω(k) = ω_0 + ω''_0 k^2 + O(k^4) and linear dispersion relation λ(ν) = d^2_{||} ν^2 + O(ν^3), using an approach that does not rely on Fenichel’s geometric singular perturbation theory.

Background

The paper establishes, via geometric singular perturbation theory, the existence of wavetrains with small spatial wavenumber k emerging from a spatially homogeneous oscillation, together with quadratic expansions of the nonlinear and linear dispersion relations. This step is crucial for analyzing the spatial spectrum and for constructing contact defects.

The authors state that they are not aware of any elementary proof of this result that avoids singular perturbation methods, highlighting a methodological gap. An elementary proof would broaden the toolkit for analyzing wavetrain bifurcations near k = 0 and potentially simplify the theory in settings where singular perturbation techniques are difficult to apply.