Journal of Nonlinear Mathematical Physics

We give a basic uniqueness theorem in the inverse spectral theory for a Sturm-Liouville equation with a weight which is not of one sign. It is shown that the theorem may be applied to the spectral problem associated with the Camassa-Holm integrable system which models shallow water waves.

It is shown that in water of finite depth, there are no periodic traveling waves with the property that the pressure in the underlying fluid flow is constant along streamlines. In the case of infinite depth, there is only one such solution, which is due to Gerstner.

We consider two-dimensional water-waves of permanent shape with a constant proagation speed. Two theorems concerning the uniqueness of certain solutions are rported. Uniqueness of Crapper's pure capillary waves is proved under a positivity assumption. The proof is based on the theory of positive operators....

The aim of this paper is to present aspects of the use of Lie groups in mechanics. We start with the motion of the rigid body for which the main concepts are extracted. In a second part, we extend the theory for an arbitrary Lie group and in a third section we apply these methods for the diffeomorphism...

We consider the direct/inverse spectral problem for the periodic Camassa-Holm eqution. In fact, we survey the direct/inverse spectral problem for the periodic weighted operator Ly = m-1 (-y +1 4 y) acting in the space L2 (R, m(x)dx), where m = uxx-u > 0 is a 1-periodic positive function and u is the...

We show that the smooth traveling waves of the Camassa-Holm equation naturally correspond to traveling waves of the Korteweg-de Vries equation.

We survey recent results on well-posedness, blow-up phenomena, lifespan and global existence for the Camassa-Holm equation. Results on weak solutions are also consiered.

The goal of this survey article is to explain the up-to-date state of the theory of Lp - Lq decay estimates for wave equations with time-dependent coefficients. We explain the influence of oscillations in the coefficients by using a precise classification. Moreover, we will see how mass and dissipation...

We prove a uniqueness result for autonomous divergence-free systems of ODE's in the plane and give an application to the study of water flows with vorticity.