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On Classification of Symmetry Reductions for the Eikonal Equation

Vasyl Fedorchuk, Volodymyr Fedorchuk
2016 Symmetry  
We study the relationship between the classification of three-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1, 4) and the types of symmetry reduction of the eikonal equation  ...  to ordinary differential equations (ODEs).  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/sym8060051 fatcat:phhnocsmhne2xhhzan26oee7nq

Symmetry Reduction and Exact Solutions of the Eikonal Equation

Ivan Fedorchuk
1996 Journal of Nonlinear Mathematical Physics  
The corresponding symmetry reduction has been done. By means of the solutions of the reduced equations some classes of exact solutions of the investigated equation have been presented.  ...  By means of splitting subgroups of the generalized Poincaré group P(1, 4), ansatzes which reduce the eikonal equation to differential equations with fewer independent variables have been constructed.  ...  The corresponding symmetry reduction has been done. Using the solutions of the reduced equations, we have found some classes of exact solutions of the eikonal equation. 1.  ... 
doi:10.2991/jnmp.1996.3.3-4.35 fatcat:ppuagtemqfaizar64cztbat73m

The symmetry and some exact solutions of the relativistic eikonal equation

W. I. Fushchich, W. M. Shtelen
1982 Lettere al Nuovo Cimento  
We discuss the symmetry group of the relativistic eikonal equation. It is found to be the conformal group C1,4 of the (4 + 1)-dimensional Poincaré-Minkowski space.  ...  Some exact multiparametrical solutions of the equation are obtained.  ...  Remarks Firstly, it id important to note that what has been said about the symmetry of eq. (2) holds true for the equation Secondly, the symmetry group of eq. (1) with m = 0 turns out to be an infinitedimensional  ... 
doi:10.1007/bf02754808 fatcat:bxoe22bcp5ay5eex5mnt33aype

On Integration of the Nonlinear d'Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Renat Z. Zhdanov
1997 Journal of Nonlinear Mathematical Physics  
We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d'Alembert equation 2u = F (u) and nonlinear eikonal equation u xµ u x µ = G(u) in the complex  ...  Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above.  ...  Saying about the d'Alembert-eikonal system, we mean the system of two nonlinear partial differential equations (PDEs) which consists of the nonlinear d'Alembert and eikonal equations taken together 2u  ... 
doi:10.2991/jnmp.1997.4.1-2.6 fatcat:rzuvuynqwrgkxclnjimntpj2g4

Symmetries of generalized soliton models and submodels on target spaceS2

C Adam, J Sanchez-Guillen
2005 Journal of High Energy Physics  
Here we calculate all the symmetries of these models and their submodels by the prolongation method.  ...  Some physically relevant non-linear models with solitons, which have target space S^2, are known to have submodels with infinitly many conservation laws defined by the eikonal equation.  ...  Further, CA acknowledges support from the Austrian START award project FWF-Y-137-TEC and from the FWF project P161 05 NO 5 of N.J. Mauser.  ... 
doi:10.1088/1126-6708/2005/01/004 fatcat:jtzpcwowzzccdnn6krrn4fyuou

Traveltime approximations for transversely isotropic media with an inhomogeneous background

Tariq Alkhalifah
2011 Geophysics  
Using perturbation theory and Taylor's series, I expand the solutions of the eikonal equation for 2D transversely isotropic media with respect to the independent parameter θ, the angle the tilt of the  ...  axis of symmetry makes with the vertical, in a generally inhomogeneous TI background with a vertical axis of symmetry (VTI).  ...  of traveltime, into the eikonal equation 3.  ... 
doi:10.1190/1.3555040 fatcat:4tzize3m7zcezl3uayl4hpxdme

Solutions of the Helmholtz equation given by solutions of the eikonal equation [article]

G.F. Torres del Castillo, Iraís Rubalcava-García
2017 arXiv   pre-print
We find the form of the refractive index such that a solution, S, of the eikonal equation yields an exact solution, ( i k_0 S), of the corresponding Helmholtz equation.  ...  Acknowledgements One of the authors (I R-G) thanks PRODEP-SEP for financial support through a postdoctoral scholarship DSA/103.5/16/5800 and also the Sistema Nacional de Investigadores (México).  ...  [1] , there exists a similar relation between solutions of the Helmholtz equation and solutions of the eikonal equation.  ... 
arXiv:1706.04120v1 fatcat:5dk44wq6dna7jdno4s6ikpc5e4

Hopf maps as static solutions of the complex eikonal equation

C. Adam
2004 Journal of Mathematical Physics  
We demonstrate that a class of torus-shaped Hopf maps with arbitrary linking number obeys the static complex eikonal equation.  ...  As this equation shows up as an integrability condition in certain non-linear field theories, the existence of such solutions is of some interest.  ...  At this point it is of interest to briefly consider the symmetries of the complex static eikonal equation (2) .  ... 
doi:10.1063/1.1792931 fatcat:lhmuqk4givdazmawmnbcro7oue

Knot soliton models, submodels, and their symmetries [article]

C. Adam, J. Sanchez-Guillen
2005 arXiv   pre-print
Here we investigate the symmetries of the submodels, where in some cases we find a symmetry enhancement for the submodels, whereas in others we do not.  ...  Further, CA acknowledges support from the Austrian START award project FWF-Y-137-TEC and from the FWF project P161 05 NO 5 of N.J. Mauser.  ...  Acknowledgments This research was partly supported by MCyT(Spain) and FEDER (FPA2002-01161), Incentivos from Xunta de Galicia and the EC network "EUCLID".  ... 
arXiv:hep-th/0511209v1 fatcat:q7olm66hd5etxmy6urkmickrom

A new traveltime approximation for TI media

Alexey Stovas, Tariq Alkhalifah
2012 Geophysics  
By expanding the TI eikonal equation in power series in terms of the aneliptic parameter η, we derive an efficient tool to estimate (scan) for η in a generally inhomogeneous, elliptically anisotropic background  ...  In all cases, the accuracy of this new TI traveltime equation exceeds previously published formulations and demonstrates how η is better resolved at small offsets when the tilt is large.  ...  ACKNOWLEDGMENTS Alexey Stovas would like to acknowledge the ROSE project for financial support. Tariq Alkhalifah is grateful to KAUST for its support.  ... 
doi:10.1190/geo2011-0158.1 fatcat:525rz3immvcplbfzkz2fsyqw5i

A hybrid eikonal solver for accurate first-arrival traveltime computation in anisotropic media with strong contrasts [article]

Kai Gao, Lianjie Huang
2020 arXiv   pre-print
We verify the stability and accuracy of our hybrid eikonal solver using several 2D and 3D TTI medium examples.  ...  We develop a stable and accurate hybrid eikonal solver for 2D and 3D transversely isotropic media with a tilted symmetry axis (TTI, or tilted transversely isotropic media).  ...  tilt angle of a VTI medium's symmetry axis (i.e., the counterclockwise rotation angle of the symmetry axis w.r.t. the y-axis), φ = φ(x) is the azimuth angle of a VTI medium's symmetry axis (i.e., the  ... 
arXiv:2008.07684v1 fatcat:j7mhkru3cjgixnhetbq3xusphi

Magnetic symmetry of geometrical optics

Miftachul Hadi
2021 Qeios  
The eikonal equation can be derived from the Maxwell equations \cite{born}.  ...  We will apply the gauge potential of the $U(1)$ magnetic symmetry to the geometrical optics, especially in the formulation of eikonal equation, the refractive index and the curvature relation.  ...  the eikonal} \cite{landau} or \textit{the optical length}, a real scalar function of position \cite{born}, then the equation (???)  ... 
doi:10.32388/gyhqx6 fatcat:2n5f7ib7zjct7icvb2llmx7gh4

An iterative, fast-sweeping-based eikonal solver for 3D tilted anisotropic media

Umair Bin Waheed, Can Evren Yarman, Garret Flagg
2015 Geophysics  
Anisotropy deviating from elliptical anisotropy introduces higher order nonlinearity into the eikonal equation, which makes solving the eikonal equation a challenge.  ...  We addressed this challenge by iteratively solving a sequence of simpler tilted elliptically anisotropic eikonal equations.  ...  ∂ x τÞ 2 ð c ∂ y τÞ 2 ð c ∂ z τÞ 2 ¼ 1: (6) The 3D tilted axis of symmetry eikonal equation For v 1 ¼ v 2 ¼ v nmo ; η 1 ¼ η 2 ¼ η; γ ¼ 1; (7) equation 6 reduces to the eikonal equation for a TTI medium  ... 
doi:10.1190/geo2014-0375.1 fatcat:ryrq5632nfhw7cmztwskbcnc6m

Pulse Propagation in Cylindrically Wound Thick Composite Specimens [chapter]

David E. Budreck
1990 Review of Progress in Quantitative Nondestructive Evaluation  
The findings, opinions, and recommendations expressed in this paper are those of the author and not necessarily those of the University of Illinois or the U.S. Navy.  ...  ACKNOWLEDGEMENTS This work was supported by the U.S. Navy through the University of Illinois (Contract #N0014-86-K-0799) and was performed by the Center for NDE at the Ames Laboratory.  ...  The eikonal equations ( 4ac) have exactly the same form as those for an orthotropic media of cartesian symmetry, where here the role of the cartesian basis ( el' e2' e3) is played by the polar basis (er  ... 
doi:10.1007/978-1-4684-5772-8_21 fatcat:7waj766yd5d5tappqj4dw4uog4

An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis

Qi Hao, Tariq Alkhalifah
2017 Geophysics  
We have developed a 2D acoustic eikonal equation governing the complex-valued traveltime of quasi P-waves in attenuating, transversely isotropic media with a vertical-symmetry axis (VTI).  ...  In addition, we have evaluated the possible extension of the proposed eikonal equation to realistic attenuating media, an alternative perturbation solution to the proposed eikonal equation, and the feasibility  ...  The main goal of this paper is to develop an acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis (VTI), in which the symmetry of the attenuation of waves  ... 
doi:10.1190/geo2016-0160.1 fatcat:ozn344dcmzdxzpkeebicgo4kse
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