Extend beyond-all-orders theory to wave (finite-wavenumber Hopf) bifurcations

Develop a general beyond-all-orders asymptotic theory for localized structures and homoclinic snaking in reaction-diffusion systems undergoing finite-wavenumber Hopf (wave) bifurcations, where the linear dispersion relation has a nonzero imaginary part; in particular, determine how to account for the coexistence of standing and travelling waves in an explicit, general calculation analogous to the Turing-case theory established in this paper.

Background

The paper develops a general beyond-all-orders asymptotic framework for homoclinic snaking near codimension-two Turing bifurcations in reaction-diffusion systems, where the dispersion curve is purely real at onset. This yields exponentially small estimates for the snaking width and associated localized structures.

In contrast, finite-wavenumber Hopf (wave) bifurcations feature a nonzero imaginary part of the dispersion relation and can produce both standing and travelling waves, significantly complicating the analysis. Although regular amplitude equations for wave bifurcations up to fifth order have been derived in prior work by the author, a full beyond-all-orders treatment comparable to the Turing case has not yet been established.

The authors explicitly identify extending the current approach to the wave-bifurcation setting as an open question, highlighting both the conceptual generalizability and the anticipated technical challenges of such a general calculation.