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

Studies on two polyherbal formulations (ZPTO and ZTO) for comparison of their antidyslipidemic, antihypertensive and endothelial modulating activities

Background Cardiovascular disorders (CVDs) are the leading cause of disease burden worldwide. Apart from available synthetic drugs used in CVDs, there are many herbal formulations including POL-10 (containing 10 herbs), which have been shown to be effective in animal studies but POL-10 was found to cause tachycardia in rodents as its side effect. This study was designed to modify the composition of POL-10 for better efficacy and/or safety profile in CVDs. Methods To assess the antidyslipidemic, antihypertensive and endothelial modulatory properties of two herbal formulations, (ZPTO and ZTO) containing Z: Zingiber officinalis, P: Piper nigrum, T: Terminalia belerica and O: Orchis mascula, different animal models including, tyloxapol and high fat diet-induced dyslipidemia and spontaneously hypertensive rats (SHR) were used. Effect on endothelial function was studied using isolated tissue bath set up coupled with PowerLab data acquisition system. The antioxidant activity was carried out using DPPH radical-scavenging assay. Results Based on preliminary screening of the ingredients of POL-10 in tyloxapol-induced hyperlipidemic rats, ZPTO and ZTO containing four active ingredients namely; Z, P, T and O were identified for further studies and comparison. In tyloxapol-induced hyperlipidemic rats, both ZPTO and ZTO caused significant reduction in serum triglyceride (TG) and total cholesterol (TC). In high fat diet-fed rats, ZPTO decreased TC, low-density lipoproteins cholesterol (LDL-C) and atherogenic index (AI). ZTO also showed similar effects to those of ZPTO with additional merits being more effective in reducing AI, body weight and more importantly raising high-density lipoproteins. In SHR, both formulations markedly reduced systolic blood pressure, AI and TG levels, ZTO being more potent in reversing endothelial dysfunction while was devoid of cardiac stimulatory effect. In addition, ZTO also reduced LDL-C and improved glucose levels in SHR. In DPPH radical-scavenging activity test, ZTO was also more potent than ZPTO. Conclusion The modified formulation, ZTO was not only found more effective in correcting cardiovascular abnormalities than ZPTO or POL-10 but also it was free from tachycardiac side-effect, which might be observed because of the presence of Piper nigrum in ZPTO

Antihypertensive, antidyslipidemic and endothelial modulating activities of Orchis mascula

The objective of this study was to investigate the possible mode(s) of action for the medicinal use of Orchis mascula (OM) (family Orchidaceae) in hypertension and dyslipidemia. In spontaneously hypertensive rats (SHRs), OM significantly (

Mutation analysis of 18 nephronophthisis associated ciliopathy disease genes using a DNA pooling and next generation sequencing strategy

Background Nephronophthisis associated ciliopathies (NPHP-AC) comprise a group of autosomal recessive cystic kidney diseases that includes nephronophthisis (NPHP), Senior-Loken syndrome (SLS), Joubert syndrome (JBTS), and Meckel-Gruber syndrome (MKS). To date, causative mutations in NPHP-AC have been described for 18 different genes, rendering mutation analysis tedious and expensive. To overcome the broad genetic locus heterogeneity, a strategy of DNA pooling with consecutive massively parallel resequencing (MPR) was devised.Methods In 120 patients with severe NPHP-AC phenotypes, five pools of genomic DNA with 24 patients each were prepared which were used as templates in order to PCR amplify all 376 exons of 18 NPHP-AC genes (NPHP1, INVS, NPHP3, NPHP4, IQCB1, CEP290, GLIS2, RPGRIP1L, NEK8, TMEM67, INPP5E, TMEM216, AHI1, ARL13B, CC2D2A, TTC21B, MKS1, and XPNPEP3). PCR products were then subjected to MPR on an Illumina Genome-Analyser and mutations were subsequently assigned to their respective mutation carrier via CEL I endonuclease based heteroduplex screening and confirmed by Sanger sequencing.Results For proof of principle, DNA from patients with known mutations was used and detection of 22 out of 24 different alleles (92% sensitivity) was demonstrated. MPR led to the molecular diagnosis in 30/120 patients (25%) and 54 pathogenic mutations (27 novel) were identified in seven different NPHP-AC genes. Additionally, in 24 patients only single heterozygous variants of unknown significance were found.Conclusions The combined approach of DNA pooling followed by MPR strongly facilitates mutation analysis in broadly heterogeneous single gene disorders. The lack of mutations in 75% of patients in this cohort indicates further extensive heterogeneity in NPHP-AC