Asymptotics of the far field generated by a modulated point source in a planarly layered electromagnetic waveguide

Barrera-Figueroa, V.; Rabinovich, Vladimir

doi:10.1002/mma.3176

Cited by 4 publications

(6 citation statements)

References 48 publications

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“…In spite of a seemingly more complicated appearance, formula becomes more convenient for computation. However, usually, the integrals appearing in it are not computed due to the obvious difficulty in calculating the solutions for a large interval with respect to ω , and G ( x , y ; λ ) is approximated by the finite sum

\sum_{j = 1}^{m} \frac{v_{j} false(x false) \overset{v_{j} false(y false)}{true}}{λ_{j} - λ}

only (see, eg, Barrera‐Figueroa et al and Barrera‐Figueroa and Rabinovich). Occasionally, in some practical models of wave propagation, such approximation can be justified with the aid of certain asymptotic considerations.…”

Section: Preliminariesmentioning

confidence: 99%

“…In practice, the computation of the Green function is performed more frequently by means of the formula (see, eg, Faddeev)

G false(x, y; λ false) = \frac{y_{1} ((), x, \sqrt{λ}) y_{2} ((), y, \sqrt{λ})}{2 i \sqrt{λ} a ((), \sqrt{λ})}, y < x,

G false(x, y; λ false) = G false(y, x; λ false),

where

λ \in double-struckC

and the branch for

\sqrt{λ}

is chosen so that

Im \sqrt{λ} \geq 0

. If

λ \neq ω_{j}^{2}

and λ ≠0 the following estimate holds

|G (x, y; λ)| \leq C e^{- Im \sqrt{λ} |x - y|} .

Often the Green function must be computed for λ being large negative numbers (see, eg, Barrera‐Figueroa et al and Barrera‐Figueroa and Rabinovich). In this case, formula presents serious numerical difficulties due to the fact that one of the solutions is exponentially increasing while the other is exponentially decreasing, in both cases with

\sqrt{λ}

participating in the exponential order.…”

Section: Preliminariesmentioning

confidence: 99%

“…Often the Green function must be computed for λ being large negative numbers (see, e.g., [2], [3]). In this case formula (7) presents serious numerical difficulties due to the fact that one of the solutions is exponentially increasing while the other is exponentially decreasing, in both cases with √ λ participating in the exponential order.…”

Section: Preliminariesmentioning

confidence: 99%

“…A method for the computation of the eigenfunctions and generalized eigenfunctions as well as of scattering data and the Green function is developed. The need for the computation of the Green function and scattering data for this operator arises, eg, in hydroacoustic problems (see, eg, Barrera‐Figueroa et al and Barrera‐Figueroa and Rabinovich) as well as when solving nonlinear evolution equation by means of the inverse scattering transform method (see Ablowitz and Clarkson).…”

Section: Introductionmentioning

confidence: 99%

“…The computation of the generalized eigenfunctions (corresponding to the continuous spectrum) is in general a complicated task. In some cases, the contribution of the discrete spectrum is considered asymptotically dominant, and consequently, the contribution of the continuous spectrum is neglected (see, eg, Barrera‐Figueroa et al and Barrera‐Figueroa and Rabinovich). In this way, an intrinsic error is introduced in the computation.…”

Section: Introductionmentioning

confidence: 99%

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A method for computation of scattering amplitudes and Green functions of whole axis problems

Castillo-Pérez

Kravchenko

Torba

2019

Math Methods in App Sciences

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A method for the computation of scattering data and of the Green function for the one‐dimensional Schrödinger operator H:=−d2dx2+qfalse(xfalse) with a decaying potential is presented. It is based on representations for the Jost solutions in the case of a compactly supported potential obtained in terms of Neumann series of Bessel functions (NSBF). The representations are used for calculating a complete orthonormal system of generalized eigenfunctions of the operator H, which in turn allow one to compute the scattering amplitudes and the Green function of the operator H−λ with λ∈double-struckC.

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