It is shown that when in a higher order variational principle one fixes fields at the boundary leaving the field derivatives unconstrained, then the variational principle (in particular the solution space) is not invariant with respect to the addition of boundary terms to the action, as it happens instead when the correct procedure is applied. Examples are considered to show how leaving derivatives of fields unconstrained affects the physical interpretation of the model. This is justified in particular...

Soient $G$ un groupe de Lie connexe de dimension $n-1$, $\Phi$ une action localement libre de classe $C^r(r\ge 2)$ de $G$ sur une variété compacte $M$ de dimension $n\ge 3$. Nous supposons qu’il existe dans l’algèbre de Lie de $G$ un champ $Y$ tel que les valeurs propres de $\mathrm{ad}\left(Y\right)$ soient ${\alpha }_1,...,{\alpha }_{n-2},0$ avec $\mathrm{Re}\left({\alpha }_i\right)\<0\forall i$. Alors, nous montrons que $\Phi$ est $C^r$-conjuguée à une “action modèle" de $G$ sur un espace homogène $H/\Gamma$ où $H$ est un groupe de Lie contenant $G$. Si $n\ge 4$, $H$ est uniquement déterminé par $G$; si $n=3$, il y a deux groupes $H$ possibles, et nous pouvons donc donner une...

We provide a detailed treatment of the Camassa-Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of a basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial...

We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in ${ℝ}^d$ to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in ${ℝ}^3$. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation,...

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma }_T$, $\mathbb{P}\left({\Sigma }_T\right)>0$ such that any data ${\phi }^{\omega }\left(x\right)\in H^{\gamma }\left({𝕋}^3\right),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-e^{it\Delta }{\phi }^{\omega }\in C([0,T];H^s\left({𝕋}^3\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1\left({𝕋}^3\right)$, that is in the supercritical scaling regime.