On periodic water waves with Coriolis effects and isobaric streamlines

In this paper we prove that solutions of the f-plane approximation for equatorial geophysical deep water waves, which have the property that the pressure is constant along the streamlines and do not possess stagnation points,are Gerstner-type waves. Furthermore, for waves traveling over a flat bed, we prove that there are only laminar flow solutions with these properties.Comment: To appear in Journal of Nonlinear Mathematical Physics; 15 page

Anomalous Creation of Branes

In certain circumstances when two branes pass through each other a third brane is produced stretching between them. We explain this phenomenon by the use of chains of dualities and the inflow of charge that is required for the absence of chiral gauge anomalies when pairs of D-branes intersect.Comment: 7 pages, two figure

On the Cauchy problem for a nonlinearly dispersive wave equation

We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite times. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.Comment: arxiv version is already officia

Variational derivation of the Camassa-Holm shallow water equation

We describe the physical hypothesis in which an approximate model of water waves is obtained. For an irrotational unidirectional shallow water flow, we derive the Camassa-Holm equation by a variational approach in the Lagrangian formalism.Comment: 10 page

On the stability of travelling waves with vorticity obtained by minimisation

We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)] to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.Comment: NoDEA Nonlinear Differential Equations and Applications, to appea

Equations of the Camassa-Holm Hierarchy

The squared eigenfunctions of the spectral problem associated with the Camassa-Holm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy, and their Hamiltonian structures, can be naturally expressed using the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. Using the GFT, we explicitly describe some members of the CH hierarchy, including integrable deformations for the CH equation. We also show that solutions of some $(1+2)$ - dimensional members of the CH hierarchy can be constructed using results for the inverse scattering transform for the CH equation. We give an example of the peakon solution of one such equation.Comment: 10 page

A Comprehensive Scan for Heterotic SU(5) GUT models

Compactifications of heterotic theories on smooth Calabi-Yau manifolds remains one of the most promising approaches to string phenomenology. In two previous papers, http://arXiv.org/abs/arXiv:1106.4804 and http://arXiv.org/abs/arXiv:1202.1757, large classes of such vacua were constructed, using sums of line bundles over complete intersection Calabi-Yau manifolds in products of projective spaces that admit smooth quotients by finite groups. A total of 10^12 different vector bundles were investigated which led to 202 SU(5) Grand Unified Theory (GUT) models. With the addition of Wilson lines, these in turn led, by a conservative counting, to 2122 heterotic standard models. In the present paper, we extend the scope of this programme and perform an exhaustive scan over the same class of models. A total of 10^40 vector bundles are analysed leading to 35,000 SU(5) GUT models. All of these compactifications have the right field content to induce low-energy models with the matter spectrum of the supersymmetric standard model, with no exotics of any kind. The detailed analysis of the resulting vast number of heterotic standard models is a substantial and ongoing task in computational algebraic geometry.Comment: 33 pages, Late