scholarly journals Dispersive Estimates for Full Dispersion KP Equations

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Didier Pilod ◽  
Jean-Claude Saut ◽  
Sigmund Selberg ◽  
Achenef Tesfahun
Keyword(s):  
Stationary Phase ◽  
Initial Value Problem ◽  
Bessel Functions ◽  
Linear Part ◽  
Independent Interest ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Initial Value ◽  
Kp Equations ◽  
Well Posed

AbstractWe prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev–Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev–Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev–Petviashvili is locally well-posed in $$H^s(\mathbb R^2)$$ H s ( R 2 ) , for $$s>\frac{7}{4}$$ s > 7 4 , in the capillary-gravity setting.

scholarly journals Focusing properties of an axicon pair

1993 ◽  
Vol 71 (1-2) ◽  
pp. 70-78 ◽  
Author(s):  
Marc Couture ◽  
Michel Piché
Keyword(s):  
Stationary Phase ◽  
Numerical Method ◽  
Gaussian Beam ◽  
Fresnel Diffraction ◽  
Optical Systems ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Diffraction Integral ◽  
Peak Intensity ◽  
Focusing Properties

The focusing properties of a so-called reflaxicon (a combination of a diverging and a converging axicon) are studied both theoretically and experimentally. Calculations of intensity distributions produced by this system are made by evaluating the Kirchhoff–Fresnel diffraction integral, first by means of an approximate technique, the stationary phase method, then by a more exact numerical method. The calculations are presented for various planes along the axis of the axicons. The effects of the presence of the supporting mount of the axicons and of some important misalignments of the system on the distributions is also investigated. Experimental results of actual intensity distributions produced by focusing a near-fundamental Gaussian beam by such a system are also presented and are seen to be in fair agreement with numerical calculations. Such calculations would be valuable in many applications for predicting important characteristics (e.g., peak intensity, length of the focal line, degree of asymmetry) of the intensity distributions formed by optical systems containing an axicon pair as the focusing component.

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

1990 ◽  
Vol 427 (1872) ◽  
pp. 221-239 ◽  
Keyword(s):  
Cauchy Problem ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Einstein Equations ◽  
Initial Value ◽  
Fluid Source ◽  
Null Hypersurfaces ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

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