We consider bifurcations of a class of infinite dimensional reversible dynamical systems which possess a family of symmetric equilibria near the origin. We also assume that the linearized operator at the origin Lε has an essential spectrum filling the entire real line, in addition to the simple eigenvalue at 0. Moreover, for parameter values ε < 0 there is a pair of imaginary eigenvalues which meet in 0 for ε = 0, and which disappear for ε > 0. The above situation occurs for example when one looks for travelling waves in a system of superposed perfect fluid layers, one being infinitely deep. We give quite general assumptions which apply in such physical examples, under which one obtains a family of bifurcating solutions homoclinic to every equilibrium near the origin. These homoclinics are symmetric and decay algebraically at infinity, being approximated at main order by the Benjamin -Ono homoclinic. For the water wave example, this corresponds to a family of solitary waves, such that at infinity the upper layer slides with a uniform velocity, over the bottom layer (at rest).

The understanding of soil organic matter (SOM) dynamics has considerably advanced in recent years. It was previously assumed that most SOM consisted of recalcitrant compounds, whereas the emerging view considers SOM as a range of polymers continuously processed into smaller molecules by decomposer enzymes. Mainstreaming this new paradigm in current models is challenging because of their ill-adapted framework. We propose the C-STABILITY model to resolve this issue. Its innovative framework combines compartmental and continuous modeling approaches to accurately reproduce SOM cycling processes. C-STABILITY emphasizes the influence of substrate accessibility on SOM turnover and makes enzymatic and microbial biotransformations of substrate explicit. Theoretical simulations provide new insights on how depolymerization and decomposers ecology impact organic matter chemistry and amount during decomposition and at steady state. The flexible mathematical structure of C-STABILITY offers a promising foundation for exploring new mechanistic hypotheses and supporting the design of future experiments.

We review the mathematical results on traveling waves in one or several superposed layers of potential flow, subject to gravity, with or without surface and interfacial tension, where the bottom layer is infinitely deep. The problem is formulated as a "spatial dynamical system," and it is shown that the linearized operator of the resulting reversible system has an essential spectrum filling the real line. We consider three cases where bifurcation occurs. (i) The first case is when, in moving a parameter, two pairs of imaginary eigenvalues merge into one pair of double eigenvalues, and then split into four symmetric complex conjugate eigenvalues. (ii) The second case is when one pair of imaginary eigenvalues meet in 0, and disappear; (iii) the third case is when the phenomenon described in (ii) is superposed to the presence of another pair of imaginary eigenvalues sitting at finite distance from 0. We give a physical example for each case and more specially study the solitary waves and generalized solitary waves, emphasizing the differences, in the methods and in the results, between these cases and the finite depth case.

No abstract