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
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 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...