A blow-up result for the periodic Camassa-Holm equation

We present a blow-up criterion for the periodic Camassa-Holm equation. The condition obtained for blow-up uses two of the conservation laws associated with the equation and improves upon some recent results.Received: 18 June 2004     

Validity of the NLS approximation for periodic quantum graphs

We consider a nonlinear Schrödinger (NLS) equation on a spatially extended periodic quantum graph. With a multiple scaling expansion, an effective amplitude equation can be derived in order to describe slow modulations in time and space of an oscillating wave packet. Using Bloch wave analysis and Gronwall’s inequality, we estimate the distance between the macroscopic approximation which is obtained via the amplitude equation and true solutions of the NLS equation on the periodic quantum graph. Moreover, we prove an approximation result for the amplitude equations which occur at the Dirac points of the system.     

Almost periodic invariant tori for the NLS on the circle

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract “counter-term theorem” à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find “many more” almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.     

A compactness result for periodic multivortices in the electroweak theory

We derive a priori uniform bounds for solutions of an elliptic system of Liouville-type equations, first analyzed by J. Spruck and Y. Yang (Comm. Math. Phys. 144 (1992) 1), yielding periodic multivortices in the classical electroweak theory of Glashow–Salam–Weinberg. Our proof is based on a concentration–quantization result, in the same spirit of Brezis–Merle (Comm. Partial Differential Equations 16 (8,9) (1991) 1223) and Li–Shafrir (Indiana Univ. Math. J. 43 (4) (1994) 1255), for mean field equations on Riemannian compact 2-manifolds.     

On invariant tori of full dimension for 1D periodic NLS

Consider the NLS with periodic boundary conditions in 1D

(0.1)     

Sharp ill-posedness result for the periodic Benjamin-Ono equation

We prove the discontinuity for the weak L²(T)-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in Hs(T) as soon as s<0 and thus completes exactly the well-posedness result obtained in Molinet (2008) [12].     

A useful result for certain linear periodic ordinary differential equations

In this paper we derive some properties for systems of linear ordinary differential equations with time-periodic coefficient matrix satisfying an additional symmetry condition (Hale's property E). These results can be used in the numerical integration of such systems and reduce the computer time by 50 %.     

Well-posedness,global existence,and blowup phenomena for a periodic quasi-linear hyperbolic equation

We establish the local well-posedness of a recently derived model for small-amplitude, shallow water waves. For a large class of initial data we prove global existence of the corresponding solution. Criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained, and an existence and uniqueness result for a class of global weak solutions is also given. © 1998 John Wiley & Sons, Inc.     

Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Bethuel et al.  and  and Chiron and Rousset [3] gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the NLS equation can approximately be described via solutions of the KdV equation associated with the wave train. Here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of ,  and . Our error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like method, and energy estimates.     

A convergence result for the periodic unfolding method related to fast diffusion on manifolds

Based on the periodic unfolding method in periodic homogenization, we deduce a convergence result for gradients of functions defined on connected, smooth, and periodic manifolds. Under the assumption of certain a-priori estimates of the gradient, which are typical for fast diffusion, the sum of a term involving a gradient with respect to the slow variable and one with respect to the fast variable is obtained in the homogenization limit. In addition, we show in a brief example how to apply this result and find for a reaction–diffusion equation defined on a periodic manifold that the homogenized equation contains a term describing macroscopic diffusion.     

A multiplicity result for periodic solutions of second order differential equations with a singularity

By the use of the Poincaré–Birkhoff fixed point theorem, we prove a multiplicity result for periodic solutions of a second order differential equation, where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity. Our main theorem is related to previous results by Rebelo (1996, 1997)  and  and Rebelo and Zanolin (1996)  and , in connection with a problem raised by del Pino et al. (1992) [1].