Co-periodic stability of periodic waves in some Hamiltonian PDEs

International audienceThe stability theory of periodic traveling waves is much less advanced than for solitary waves, which were first studied by Boussinesq and have received a lot of attention in the last decades. In particular, despite recent breakthroughs regarding periodic waves in reaction-diffusion equations and viscous systems of conservation laws [Johnson–Noble–Rodrigues–Zumbrun, Invent math (2014)], the stability of periodic traveling wave solutions to dispersive PDEs with respect to 'arbitrary' perturbations is still widely open in the absence of a dissipation mechanism. The focus is put here on co-periodic stability of periodic waves, that is, stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Fairly general nonlinearities are allowed in these systems, so as to include various models of mathematical physics, and this precludes complete integrability techniques. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg–de Vries equation (qKdV), and the Euler–Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability , and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. However, the resulting criteria are more complicated for periodic waves because they have more degrees of freedom than solitary waves, so that the action is a function of N + 2 variables for a system of N PDEs, while the moment of instability is a function of the wave speed only once the endstate of the 1 solitary wave is fixed. Regarding solitary waves, the celebrated Grillakis–Shatah– Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, stability criteria involve all the second order, partial derivatives of the action. This had already been pointed out by various authors for some specific equations, in particular the generalized Korteweg–de Vries equation — which is special case of (qKdV) — but not from a general point of view, up to the authors' knowledge. The most striking results obtained here can be summarized as: an odd value for the difference between N and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Furthermore, it is shown that, when applied to the Euler–Korteweg system, this approach yields several interesting connexions between (EKE), (EKL), and (qKdV). More precisely, (EKE) and (EKL) share the same abbreviated action integral, which is related to that of (qKdV) in a simple way. This basically proves simultaneous stability in both formulations (EKE) and (EKL) — as one may reasonably expect from the physical point view —, which is interesting to know when these models are used for different phenomena — e.g. shallow water waves or nonlinear optics. In addition, stability in (EKE) and (EKL) is found to be linked to stability in the scalar equation (qKdV). Since the relevant stability criteria are merely encoded by the negative signature of (N + 2) × (N + 2) matrices, they can at least be checked numerically. In practice, when N = 1 or 2, this can be done without even requiring an ODE solver. Various numerical experiments are presented, which clearly discriminate between stable cases and unstable cases for (qKdV), (EKE) and (EKL), thus confirming some known results for the generalized KdV equation and the Nonlinear Schrödinger equation, and pointing out some new results for more general (systems of) PDEs

Analyses and localization of pectin-like carbohydrates in cell wall and mucilage of the green alga Netrium digitus

The unicellular, simply shaped desmid Netrium digitus inhabiting acid bog ponds grows in two phases. Prior to division, the cell elongates at its central zone, whereas in a second phase, polar tip growth occurs. Electron microscopy demonstrates that Netrium is surrounded by a morphologically homogeneous cell wall, which lacks pores. Immunocytochemical and biochemical analyses give insight into physical wall properties and, thus, into adaptation to the extreme environment. The monoclonal antibodies JIM5 and JIM7 directed against pectic epitopes with different degrees of esterification label preferentially growing wall zones in Netrium. In contrast, 2F4 marks the cell wall only after experimental de-esterification. Electron energy loss spectroscopy reveals Ca-binding capacities of pectins and gives indirect evidence for the degree of their esterification. An antibody raised against Netrium mucilage is not only specific to mucilage but also recognizes wall components in transmission electron microscopy and dot blots. These results indicate a smooth transition between mucilage and the cell wall in Netrium

Computing the Maslov index of solitary waves, Part 2: Phase space with dimension greater than four

This paper extends the theory of the Maslov index of solitary waves in Part 1 to the case where the phase space is of dimension greater than four. The starting point is Hamiltonian PDEs, in one space dimension and time, whose steady part is a Hamiltonian ODE with a phase space of dimension six or greater. This steady Hamiltonian ODE is the main focus of the paper. Homoclinic orbits of the steady ODE represent solitary waves of the PDE, and one of the properties of the homoclinic orbits is the Maslov index. We develop formulae for the Maslov index, the Maslov angle and its subangles, in an exterior algebra framework, and develop numerical algorithms to compute them. In addition, a new numerical approach based on a discrete QR algorithm is proposed. The Maslov index is of interest for classifying solitary waves and as an indicator of stability or instability of the solitary wave in the time-dependent problem. The theory is applied to a class of reaction–diffusion equations, the longwave–shortwave resonance equations and the seventh-order KdV equation

Do African grey parrots (Psittacus erithacus) know what a human experimenter does and does not see?

Perspective-taking is a cognitive ability that can be useful to access information during social interactions. This ability is extensively exploited in humans, and some evidence of it has been found in other mammals and some bird species. Perspective-taking requires individuals to be sensitive to the attentional state of others. In this experiment, three hand-reared grey parrots were tested on their ability to adapt their behaviours according to the perception of a human handler. Two different screens placed on a table separating the human side from the parrot's side were used: one transparent and one opaque. In the Control condition food was put behind each screen, whereas in the Test condition ‘forbidden’ objects (attractive for the bird but normally not accessible) were placed behind each screen. Birds were expected to choose at random between the two screens in the Control condition but to prefer the opaque one in the Test condition in order to avoid being scolded and chased away. In the Control condition, birds chose at random, whereas the older parrot chose the opaque screen significantly more in the Test condition. The latency for the decision was longer in the Test condition compared to Control, and when choosing the Transparent screen compared to the Opaque