Journal of Nonlinear Mathematical Physics

Volume 11, Issue 4, November 2004
List of participants

1. On the Spectral Problem Associated with the Camassa-Holm Equation

Pages: 422 - 432
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.

3. Periodic Traveling Water Waves with Isobaric Streamlines

Pages: 461 - 471
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.

4. Uniqueness Issues on Permanent Progressive Water-Waves

Pages: 472 - 479
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....

5. Lie Groups and Mechanics: An Introduction

Pages: 480 - 498
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...

6. Inverse Spectral Problem for the Periodic Camassa-Holm Equation

Pages: 499 - 507
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...

7. Traveling Wave Solutions of the Camassa-Holm and Korteweg-de Vries Equations

Pages: 508 - 520
We show that the smooth traveling waves of the Camassa-Holm equation naturally correspond to traveling waves of the Korteweg-de Vries equation.

8. On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey

Pages: 521 - 533
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.

9. Lp - Lq Decay Estimates for Wave Equations with Time-Dependent Coefficients

Pages: 534 - 548
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...

10. Uniqueness for Autonomous Planar Differential Equations and the Lagrangian Formulation of Water Flows with Vorticity

Pages: 549 - 555
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.