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A short note on the asymptotic stability of an oscillation-free eikonal splitting method

Qin Sheng, Shekhar Guha, Leonel P. Gonzalez
2012 Applied Mathematics Letters  
It is shown that the finite difference method investigated is not only oscillation-free, but also asymptotically stable.  ...  This paper concerns an eikonal splitting BPM scheme for two-dimensional paraxial Helmholtz equations together with transparent boundary conditions in slowly varying envelope approximations of active laser  ...  This study is supported in part by a research grant (No. F-5400-04-06-SC01-00) from the Air Force Research Laboratory and General Dynamics Information Technology.  ... 
doi:10.1016/j.aml.2012.01.011 fatcat:f2zldzqyffdtfn7wshhg6zatha

Asymptotic Stability of an Eikonal Transformation Based ADI Method for the Paraxial Helmholtz Equation at High Wave Numbers

Qin Sheng, Hai-Wei Sun
2012 Communications in Computational Physics  
AbstractThis paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation  ...  Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense.  ...  The study of the first author was supported in part by a URC Award from the Baylor University. The second author appreciates very much the University of Macau for the research grant (No.  ... 
doi:10.4208/cicp.100811.090112a fatcat:swwsccdvlvbvlirh5wpph35tka

Exponential splitting for n -dimensional paraxial Helmholtz equation with high wavenumbers

Qin Sheng, Hai-Wei Sun
2014 Computers and Mathematics with Applications  
An eikonal transformation is introduced for oscillation-free platforms and matrix operator decompositions.  ...  This paper explores applications of the exponential splitting method for approximating highly oscillatory solutions of the n-dimensional paraxial Helmholtz equation.  ...  The second author also appreciates the support and encouragements from the Science and Technology Development Fund and the University of Macau, Macao, China.  ... 
doi:10.1016/j.camwa.2014.09.005 fatcat:6ms3rzj37jganlokyt6ciuloia

About nonlinear geometric optics [article]

Eric Dumas
2005 arXiv   pre-print
We give an idea of the evolution of mathematical nonlinear geometric optics from its foundation by Lax in 1957, and present applications in various fields of mathematics and physics.  ...  In [Chear] , he converts the stability problem into a "cascade of phases" phenomenon, where the phase has an asymptotic expansion, whose coefficients are coupled with amplitudes.  ...  approximately solution to the equation); 2-showing that for any initial data close to the initial value of the approximate solution, an exact solution exists on a time interval independent of ε; 3(stability  ... 
arXiv:math/0511629v1 fatcat:pamcv46zkvaifkfwd3lgndekjq

Analytical solutions for strong field-driven atomic and molecular one- and two-electron continua and applications to strong-field problems

Olga Smirnova, Michael Spanner, Misha Ivanov
2008 Physical Review A. Atomic, Molecular, and Optical Physics  
The main component of this approach is the approximate solution for the continuum states of one and two electrons in the presence of a long-range ionic potential and a strong laser field.  ...  The large-angle scattering ͑hard collision͒ is incorporated in the EVA formalism using an expansion in the number of hard collisions.  ...  M.I., M.S., and O.S. acknowledge financial support by the NSERC SRO. M.I. also acknowledges support from the Alexander von Humboldt Foundation.  ... 
doi:10.1103/physreva.77.033407 fatcat:ip6gg2wcubewpibl4qsbjjeerq

Coherent Structures Generated by Inhomogeneities in Oscillatory Media

Richard Kollár, Arnd Scheel
2007 SIAM Journal on Applied Dynamical Systems  
We investigate the effect of spatially localized inhomogeneities on a spatially homogeneous oscillation in a reaction-diffusion system.  ...  We also derive asymptotics for wavenumbers and group velocities in the far field.  ...  We conclude with a discussion in Section 4, including a summary of results, discussion of topological and quantitative classifications of defects, and a note on asymptotic stability.  ... 
doi:10.1137/060666950 fatcat:kmjoerrjybbq5d4rt5ukpogiyy

Coherent Structures Generated by Inhomogeneities in Oscillatory Media

Richard Kollár, Arnd Scheel
2008 SIAM Journal of Imaging Sciences  
We investigate the effect of spatially localized inhomogeneities on a spatially homogeneous oscillation in a reaction-diffusion system.  ...  We also derive asymptotics for wavenumbers and group velocities in the far field.  ...  We conclude with a discussion in Section 4, including a summary of results, discussion of topological and quantitative classifications of defects, and a note on asymptotic stability.  ... 
doi:10.1137/test8 fatcat:5yvee2f5sfhrvkrpnlogjlnyle

Magnetosphere of a Kerr black hole immersed in magnetized plasma and its perturbative mode structure

Huan Yang, Fan Zhang, Luis Lehner
2015 Physical Review D  
In the second part of this work, we analyze the perturbations of this family of jet-like solutions, and show that the perturbative modes exhibit a similar split into the trapped and traveling categories  ...  While these solutions all satisfy the force-free equations and the appropriate boundary conditions, our numerical experiments show a unique relaxed state starting from different initial data, and so one  ...  Finally, we note that although the discussion in this section relies on an expansion in a, we expect this basic picture of a tube dividing the magnetosphere into a jet region and an outside region to remain  ... 
doi:10.1103/physrevd.91.124055 fatcat:t76hethqfreohpndhrwaxza2hu

Asymptotic Theory For Diffusive Electromagnetic Imaging

Jean Virieux, Carlos Flores-Luna, Dominique Gibert
1994 Geophysical Journal International  
of the electric signal obtained at the free surface. (3) This linear relation is asymptotically inverted by using an iterative quasi-Newtonian inversion based on a least-squares criterion developed by  ...  S U M M A R Y W e propose an asymptotic theory for diffusive electromagnetic imaging.  ...  ACKNOWLEDGMENTS This work was partly supported by the CNRS-INSU through the 'Tomographic' group and MEN-DRED through Jeune Equipe 'RuaDE'. We thank two anonymous reviewers for their helpful comments.  ... 
doi:10.1111/j.1365-246x.1994.tb04022.x fatcat:fi6cb7wntbaxjjdo5ajvovglda

Theory of near-field matter-wave interference beyond the eikonal approximation

Stefan Nimmrichter, Klaus Hornberger
2008 Physical Review A. Atomic, Molecular, and Optical Physics  
It serves to systematically improve the eikonal approximation used so far, and to assess its regime of validity.  ...  We provide a general description of the grating interaction process by combining semiclassical scattering theory with a phase space formulation.  ...  The latter defines the asymptotic final position r 2T of the trajectory by means of a free evolution determined by the asymptotic final momentum p 2T .  ... 
doi:10.1103/physreva.78.023612 fatcat:iuuvzfgatjbizjya552ubbmvzy

PROGRESS IN ASYMMETRIC RESONANT CAVITIES: USING SHAPE AS A DESIGN PARAMETER IN DIELECTRIC MICROCAVITY LASERS [chapter]

H. G. L. Schwefel, H. E. Tureci, A. Douglas Stone, R. K. Chang
2004 Optical Microcavities  
Theoretical analysis of the directional emission from polymer lasers has shown that highly directional emission is compatible with strongly chaotic ray dynamics due to the non-random character of the short-term  ...  Lasing modes based on stable and unstable periodic orbits have been found as well as modes based on chaotic whispering gallery orbits; the lasing mode depending on the material, shape and index of refraction  ...  The eikonal method consists of attempting an asymptotic solution of Eq. (1) (now with n = 1) of the form ψ(x, y) = A(x, y)e ikS(x,y) where kR 1 and S,A are real functions independent of k and AA 0 is  ... 
doi:10.1142/9789812565730_0009 fatcat:5qmcosseavg35okpc43svjlu7u

Asymptotic theory of gravity modes in rotating stars

V Prat, S Mathis, K Augustson, F Lignières, J Ballot, L Alvan, A. S Brun
2018 Astronomy and Astrophysics  
We investigate the impact of a general differential rotation (both in radius and latitude) on the propagation of axisymmetric gravito-inertial waves. Methods.  ...  Some parts of oscillation spectra corresponding to regimes similar to those of the uniformly rotating case would exhibit regular patterns, while parts corresponding to the new regimes would be mostly constituted  ...  The authors also thank the referee for giving useful comments that have allowed us to improve the article.  ... 
doi:10.1051/0004-6361/201832576 fatcat:ylsywkmsy5e5nja26ggm4shbxu

Fundamental problems of modeling the dynamics of internal gravity waves with applications to the Arctic Basin [article]

Vitaly V. Bulatov Institute for Problems in Mechanics of the Russian Academy of Sciences)
2012 arXiv   pre-print
The numerical and analytical results show that the internal gravity waves have a significant effect on underwater sea objects in the Arctic Basin.  ...  We present analytical and numerical algorithms for calculating the wave fields for a set of values of the parameters, as observed in the ocean.  ...  ACKNOWLEDGMENTS The results presented in the paper have been obtained by research performed under projects supported by the Russian Foundation for Basic Research (No.11-01-00335а "Stability and Instability  ... 
arXiv:1206.6136v1 fatcat:wfcwn3uep5gqvdwydgqpri7naa

Front dynamics in an oscillatory bistable Belousov-Zhabotinsky chemical reaction

Bradley Marts, Karl Martinez, Anna L. Lin
2004 Physical Review E  
We measure the curvaturedriven shrinking of a circular domain R ϳ t 1/2 at forcing frequencies below a specific value, and show that the fast time scale front oscillations (breathing) drive this slow time  ...  Just below the transition frequency is a relatively narrow range of frequencies where the curvature-driven coarsening is balanced by a competing front interaction, which leads to a pattern of localized  ...  The radius after expansion is r 1/2 = ͱ A 0 + ⌬A = ͱ r 0 2 + 2r 0 ⌬x, ͑4͒ where r 1/2 is the radius of the circle after one half oscillation. Note that r 1/2 is less than r 0 + ⌬x.  ... 
doi:10.1103/physreve.70.056223 pmid:15600744 fatcat:gfsmlflxr5glplt6rjuwjcsfu4

Mathematical and computational methods for semiclassical Schrödinger equations

Shi Jin, Peter Markowich, Christof Sparber
2011 Acta Numerica  
Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical  ...  The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge.  ...  ., an asymptotic resolution of the oscillation is not always enough.  ... 
doi:10.1017/s0962492911000031 fatcat:osty3kus3ngcdkpomkovrdxhou
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