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

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

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

Particle trajectories in linearized irrotational shallow water flows

We investigate the particle trajectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water's free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending on the strength of underlying uniform current, we obtain that some particle trajectories are undulating path to the right or to the left, some are looping curves with a drift to the right and others are parabolic curves or curves which have only one loop

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

Adversarially Tuned Scene Generation

Generalization performance of trained computer vision systems that use computer graphics (CG) generated data is not yet effective due to the concept of 'domain-shift' between virtual and real data. Although simulated data augmented with a few real world samples has been shown to mitigate domain shift and improve transferability of trained models, guiding or bootstrapping the virtual data generation with the distributions learnt from target real world domain is desired, especially in the fields where annotating even few real images is laborious (such as semantic labeling, and intrinsic images etc.). In order to address this problem in an unsupervised manner, our work combines recent advances in CG (which aims to generate stochastic scene layouts coupled with large collections of 3D object models) and generative adversarial training (which aims train generative models by measuring discrepancy between generated and real data in terms of their separability in the space of a deep discriminatively-trained classifier). Our method uses iterative estimation of the posterior density of prior distributions for a generative graphical model. This is done within a rejection sampling framework. Initially, we assume uniform distributions as priors on the parameters of a scene described by a generative graphical model. As iterations proceed the prior distributions get updated to distributions that are closer to the (unknown) distributions of target data. We demonstrate the utility of adversarially tuned scene generation on two real-world benchmark datasets (CityScapes and CamVid) for traffic scene semantic labeling with a deep convolutional net (DeepLab). We realized performance improvements by 2.28 and 3.14 points (using the IoU metric) between the DeepLab models trained on simulated sets prepared from the scene generation models before and after tuning to CityScapes and CamVid respectively.Comment: 9 pages, accepted at CVPR 201