Do peaked solitary water waves indeed exist?

Many models of shallow water waves admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa-Holm equation. Besides, it is proved that Kelvin's theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, the phase speed of the peaked solitary waves has nothing to do with wave height. In addition, the kinetic energy of the peaked solitary waves either increases or almost keeps the same from free surface to bottom. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.Comment: 53 pages, 13 figures, 7 tables. Accepted by Communications in Nonlinear Science and Numerical Simulatio

Optimization of Renormalization Group Flow

Renormalization group flow equations for scalar lambda Phi^4 are generated using three classes of smooth smearing functions. Numerical results for the critical exponent nu in three dimensions are calculated by means of a truncated series expansion of the blocked potential. We demonstrate how the convergence of nu as a function of the order of truncation can be improved through a fine tuning of the smoothness of the smearing functions.Comment: 23 pages, 7 figure

Finding the one-loop soliton solution of the short-pulse equation by means of the homotopy analysis method

The homotopy analysis method is applied to the short-pulse equation in order to find an analytic approximation to the known exact solitary upright-loop solution. It is demonstrated that the approximate solution agrees well with the exact solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves

Statistical properties of solutions to the Navier-Stokes equation in the limit of vanishing viscosity

Presents a new approach to the Navier-Stokes turbulence. With the Gaussian soft constraint on the Navier-Stokes equation, the author derives the energy spectrum to be E(k) approximately k^-3 and k^-2 in two and three spatial dimensions respectively. The possible future developments are also pointed out

Book Review: Understanding commercial Law

This article reviews the book: “Understanding commercial Law”, by Philippa Gerbic and Leigh Miller

Hard Probe of Soft Matter Geometry and Fluctuations from RHIC to LHC

We report results on event-by-event hard probe of soft matter geometry and fluctuations in heavy ion collisions. Geometric data ($v_2$ of high $p_t$ hadrons) from RHIC plus LHC clearly favors jet "monography" model with strong near-Tc enhancement of jet-medium interaction strength which also implies a less opaque medium at LHC. We also quantify the jet responses to all harmonic anisotropy $v_n$($n=1,2,3,4,5,6$) and their manifestation in hard-soft azimuthal correlations.Comment: 3 pages, 4 figures, contribution to CIPANP2012 Proceeding

Book Review: Hinde on Commercial Leases

This article reviews the book: “Hinde on Commercial Leases”, by G.W. Hinde