Nonlinear Partial Differential Equations

DiBenedetto, Emmanuele

doi:10.1090/conm/238

Contemporary Mathematics

1999

DOI: 10.1090/conm/238

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Nonlinear Partial Differential Equations

Emmanuele DiBenedetto¹

Abstract: The conformal Galilei algebra (cga) and the exotic conformal Galilei algebra (ecga) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the cga but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under cga.

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Cited by 3 publications

References 11 publications

(28 reference statements)

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Decay estimates of solutions to the bipolar non-isentropic compressible Euler–Maxwell system

2017

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We study the large time behavior of solutions near a constant equilibrium to the compressible Euler-Maxwell system in R 3 . We first refine a global existence theorem by assuming that the H 3 norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If the initial data belongs toḢ −s (0 ≤ s < 3/2) orḂ −s 2,∞ (0 < s ≤ 3/2), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the usual L p -L 2 (1 ≤ p ≤ 2) type of the decay rates follow without requiring that the L p norm of initial data is small.2010 Mathematics Subject Classification. 83C22; 82D37; 76N10; 35Q35; 35B40.

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Decay estimates of solutions to the bipolar non-isentropic compressible Euler–Maxwell system

2017

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Local algorithm for computing complex travel time based on the complex eikonal equation

Huang

Sun

2016

Phys. Rev. E

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The traditional algorithm for computing the complex travel time, e.g., dynamic ray tracing method, is based on the paraxial ray approximation, which exploits the second-order Taylor expansion. Consequently, the computed results are strongly dependent on the width of the ray tube and, in regions with dramatic velocity variations, it is difficult for the method to account for the velocity variations. When solving the complex eikonal equation, the paraxial ray approximation can be avoided and no second-order Taylor expansion is required. However, this process is time consuming. In this case, we may replace the global computation of the whole model with local computation by taking both sides of the ray as curved boundaries of the evanescent wave. For a given ray, the imaginary part of the complex travel time should be zero on the central ray. To satisfy this condition, the central ray should be taken as a curved boundary. We propose a nonuniform grid-based finite difference scheme to solve the curved boundary problem. In addition, we apply the limited-memory Broyden-Fletcher-Goldfarb-Shanno technology for obtaining the imaginary slowness used to compute the complex travel time. The numerical experiments show that the proposed method is accurate. We examine the effectiveness of the algorithm for the complex travel time by comparing the results with those from the dynamic ray tracing method and the Gauss-Newton Conjugate Gradient fast marching method.

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A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

Huang

Sun

et al. 2020

GEOPHYSICS

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The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have developed analytic approximations for the complex traveltime in transversely isotropic media with a titled symmetry axis, which is defined by a Taylor series expansion over the anisotropy parameters. The formulation for the complex traveltime is developed using perturbation theory and the complex point-source method. The real part of the complex traveltime describes the wavefront, and the imaginary part of the complex traveltime describes the decay of the amplitude of waves away from the central ray. We derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion using perturbation theory. The analytical solutions for the complex traveltimes are determined by applying the complex point-source method to the background traveltime formula and subsequently obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods for benchmarking approximated solutions.

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Nonlinear Partial Differential Equations

Cited by 3 publications

References 11 publications

Decay estimates of solutions to the bipolar non-isentropic compressible Euler–Maxwell system

Decay estimates of solutions to the bipolar non-isentropic compressible Euler–Maxwell system

Local algorithm for computing complex travel time based on the complex eikonal equation

A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

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