A Kam Theorem for Space-Multidimensional Hamiltonian PDE

We present an abstract KAM theorem, adapted to space-multidimensional hamiltonian PDEs with smoothing non-linearities. The main novelties of this theorem are that: $\bullet$ the integrable part of the hamiltonian may contain a hyperbolic part and as a consequence the constructed invariant tori may be unstable. $\bullet$ It applies to singular perturbation problem. In this paper we state the KAM-theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem .Comment: arXiv admin note: text overlap with arXiv:1502.0226

Clinical presentation of calmodulin mutations: the International Calmodulinopathy Registry

AIMS: Calmodulinopathy due to mutations in any of the three CALM genes (CALM1-3) causes life-threatening arrhythmia syndromes, especially in young individuals. The International Calmodulinopathy Registry (ICalmR) aims to define and link the increasing complexity of the clinical presentation to the underlying molecular mechanisms. METHODS AND RESULTS: The ICalmR is an international, collaborative, observational study, assembling and analysing clinical and genetic data on CALM-positive patients. The ICalmR has enrolled 140 subjects (median age 10.8 years [interquartile range 5-19]), 97 index cases and 43 family members. CALM-LQTS and CALM-CPVT are the prevalent phenotypes. Primary neurological manifestations, unrelated to post-anoxic sequelae, manifested in 20 patients. Calmodulinopathy remains associated with a high arrhythmic event rate (symptomatic patients, n = 103, 74%). However, compared with the original 2019 cohort, there was a reduced frequency and severity of all cardiac events (61% vs. 85%; P = .001) and sudden death (9% vs. 27%; P = .008). Data on therapy do not allow definitive recommendations. Cardiac structural abnormalities, either cardiomyopathy or congenital heart defects, are present in 30% of patients, mainly CALM-LQTS, and lethal cases of heart failure have occurred. The number of familial cases and of families with strikingly different phenotypes is increasing. CONCLUSION: Calmodulinopathy has pleiotropic presentations, from channelopathy to syndromic forms. Clinical severity ranges from the early onset of life-threatening arrhythmias to the absence of symptoms, and the percentage of milder and familial forms is increasing. There are no hard data to guide therapy, and current management includes pharmacological and surgical antiadrenergic interventions with sodium channel blockers often accompanied by an implantable cardioverter-defibrillator

Four lectures on KAM for the non-linear Schrödinger equation

International audienceWe discuss the KAM-theory for lower-dimensional tori for the non-linear Schrödinger equation with periodic boundary conditions and a convolution potential in dimension d. Central in this theory is the homological equation and a condition on the small divisors often known as the second Melnikov condition. The difficulties related to this condition are substantial when d≥ 2. We discuss this difficulty, and we show that a block decomposition and a Töplitz- Lipschitz-property, present for non-linear Schrödinger equation, permit to overcome this difficuly. A detailed proof is given in [EK06]

KAM for the Non-Linear Schrödinger Equation

International audienceWe consider the $d$-dimensional nonlinear Schrödinger equation under periodic boundary conditions: -i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d where V(x)=\sum \hat V(a)e^{i\sc{a,x}} is an analytic function with $\hat V$ real, and $F$ is a real analytic function in $\Re u$, $\Im u$ and $x$. (This equation is a popular model for the 'real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For \ep=0 the equation is linear and has time--quasi-periodic solutions $u$, u(t,x)=\sum_{a\in Å}\hat u(a)e^{i(|a|^2+\hat V(a))t}e^{i\sc{a,x}} \quad (|\hat u(a)|>0), where $Å$ is any finite subset of $\Z^d$. We shall treat $\omega_a=|a|^2+\hat V(a)$, $a\inÅ$, as free parameters in some domain $U\subset\R^{Å}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If |\ep| is sufficiently small, then there is a large subset $U'$ of $U$ such that for all $\omega\in U'$ the solution $u$ persists as a time--quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients

KAM for the non-linear Schrödinger equation - A short presentation

International audienc