On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations

Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given.Comment: 26 page

Effective good divisibility of rational homogeneous varieties

We compute the effective good divisibility of a rational homogeneous variety, extending an earlier result for complex Grassmannians by Naldi and Occhetta. Non-existence of nonconstant morphisms to rational homogeneous varieties of classical Lie type are obtained as applications.Comment: 22 pages. 2 figures. Comments are welcom

Massive Thirring Model: Inverse Scattering and Soliton Resolution

In this paper the long-time dynamics of the massive Thirring model is investigated. Firstly the nonlinear steepest descent method for Riemann-Hilbert problem is explored to obtain the soliton resolution of the solutions to the massive Thirring model whose initial data belong to some weighted-Sobolev spaces. Secondly, the asymptotic stability of multi-solitons follow as a corollary. The main difficulty in studying the massive Thirring model through inverse scattering is that the corresponding Lax pair has singularities at the origin and infinity. We overcome this difficulty by making use of two transforms that separate the singularities.Comment: arXiv admin note: text overlap with arXiv:2009.04260, arXiv:1907.0711

Talbot effect for the Manakov System on the torus

In this paper, the Talbot effect for the multi-component linear and nonlinear systems of the dispersive evolution equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles is investigated. Firstly, for a class of two-component linear systems satisfying the dispersive quantization conditions, we discuss the fractal solutions at irrational times. Next, the investigation to nonlinear regime is extended, we prove that, for the concrete example of the Manakov system, the solutions of the corresponding periodic initial-boundary value problem subject to initial data of bounded variation are continuous but nowhere differentiable fractal-like curve with Minkowski dimension $3/2$ at irrational times. Finally, numerical experiments for the periodic initial-boundary value problem of the Manakov system, are used to justify how such effects persist into the multi-component nonlinear regime. Furthermore, it is shown in the nonlinear multi-component regime that the interplay of different components may induce subtle different qualitative profile between the jump discontinuities, especially in the case that two nonlinearly coupled components start with different initial profile

Dynamic characteristic of spur gear with flexible support of gearbox

In this study, a nonlinear translation-torsion model of spur gear pair with flexible support of gearbox is proposed. The time-varying meshing stiffness, transmission error and backlash are considered in this model. Lagrange’s equations are used for establishing the mathematic model. The numerical method is presented for solutions of nonlinear differential equations. The effect of rotating speed and support stiffness of gearbox is analyzed. The numerical results show that the flexibility of the support of gearbox has a significant effect on the amplitude-frequency characteristic of the spur gear pair at low rotating speeds. The response shows flexibility while the support stiffness is smaller than the bearings and rigidity while the support stiffness is larger than the bearings. The maximum deformation of the driving gear bearings under the flexible support is generally greater than the one under rigid support

Simpler is Better: Predicting Consumer Vehicle Purchases in the Short Run

When agencies such as the US Environmental Protection Agency (EPA) establish future greenhouse gas emissions standards for new vehicles, forecasting future vehicle purchases due to changes in fuel economy and prices provides insight into regulatory impacts. We compare predictions from a nested logit model independently developed for US EPA to a simple model where past market share predicts future market share using data from model years 2008, 2010, and 2016. The simple model outperforms the nested logit model for all goodness-of-prediction measures for both prediction years. Including changes in vehicle price and fuel economy increases bias in forecasted market shares. This bias suggests price increases are correlated with unobserved increases in vehicle quality, changes in preferences, or brand-specific changes in market size but not cost pass-through. For 2010, past shares predict better than a nested logit model despite a major shock, the economic disruption caused by the Great Recession. Observed share changes during this turbulent period may offer upper bounds for policy changes in other contexts: the largest observed change in market share across the two horizons is 6.6% for manufacturers in 2016 and 3.4% for an individual vehicle in 2010