In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities....

We survey some recent results for the KP-II equation. We also give an idea for treating the “bad frequency interactions” of the bilinear estimates in the Fourier transform restriction spaces related to the KP-I equation.

These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation $u_t+{(u_{xx}+u^5)}_x=0$ for initial data in $H^1$ close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in $H^1$, construction of various exotic blow up rates in $H^1$, including grow up in infinite time.

We consider the mass critical (gKdV) equation $u_t+{(u_{xx}+u^5)}_x=0$ for initial data in $H^1$. We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

Si considera il problema di Cauchy per l'equazione (cf. [1]): $${\varphi }_{tt}-{\varphi }_{xx}-{\varphi }_x^2{\varphi }_{xx}+\sin \varphi =0\left(x,t\right)\in \mathbb{R}\times {\mathbb{R}}_{+}.$$ Nella prima parte di questo articolo si dimostra, per dati iniziali particolari, un risultato di «blow-up» della soluzione classica locale (in tempo), seguendo le idee introdotte in [8], [2] ed [4]. Nella seconda parte, viene utilizzato il metodo di compattezza per compensazione (cf. [13], [10] ed [5]) ed una estensione del principio delle regioni invarianti (cf. [12]) per dimostrare l'esistenza di una soluzione debole globale entropica....

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s\ge -\frac{1}{4}$ (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in $H^{-1}$ (joint work with T. Buckmaster [2]).

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal. 178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal.178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...

We investigate the long-time behaviour of solutions to the Korteweg-de Vries equation with a zero order dissipation and an additional forcing term, when the space variable varies over $R$, and prove that it is described by a maximal compact attractor in $H^2\left(R\right)$.

Partant du principe de conservation de la masse et du principe fondamental de la dynamique, on retrouve l'équation d'Euler nous permettant de décrire les modèles asymptotiques de propagation d'ondes dans des eaux peu profondes en dimension 1. Pour décrire la propagation des ondes en dimension 2, Kadomtsev et Petviashvili [ 15 (1970) 539] utilisent une perturbation linéaire de l'équation de KdV. Mais cela ne précise pas si les équations ainsi obtenues dérivent de l'équation d'Euler, c'est ce que...