Abstract
The inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered. In the limiting process we find that the widths of the forbidden bands approach the amplitudes of the Fourier spectrum. The number of spectral bands is estimated from Fourier theory and depends explicitly on the assumed spatial discretization in the wave amplitude function (potential). This allows one to estimate the number of degrees of freedom in a discrete (and, therefore, finite-banded) potential. An essential feature of the calculations is that all results for the periodic problem are cast in terms of the infinite-line reflection and transmission coefficients b(k), a(k). Thus the connection between the whole-line and periodic problems is clear at every stage of the computations.
Citation Formats
Osborne, A R, and Bergamasco, L.
Small-amplitude limit of the spectral transform for the periodic Korteweg-de Vries equation.
Italy: N. p.,
1985.
Web.
doi:10.1007/BF02721563.
Osborne, A R, & Bergamasco, L.
Small-amplitude limit of the spectral transform for the periodic Korteweg-de Vries equation.
Italy.
https://doi.org/10.1007/BF02721563
Osborne, A R, and Bergamasco, L.
1985.
"Small-amplitude limit of the spectral transform for the periodic Korteweg-de Vries equation."
Italy.
https://doi.org/10.1007/BF02721563.
@misc{etde_5366791,
title = {Small-amplitude limit of the spectral transform for the periodic Korteweg-de Vries equation}
author = {Osborne, A R, and Bergamasco, L}
abstractNote = {The inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered. In the limiting process we find that the widths of the forbidden bands approach the amplitudes of the Fourier spectrum. The number of spectral bands is estimated from Fourier theory and depends explicitly on the assumed spatial discretization in the wave amplitude function (potential). This allows one to estimate the number of degrees of freedom in a discrete (and, therefore, finite-banded) potential. An essential feature of the calculations is that all results for the periodic problem are cast in terms of the infinite-line reflection and transmission coefficients b(k), a(k). Thus the connection between the whole-line and periodic problems is clear at every stage of the computations.}
doi = {10.1007/BF02721563}
journal = []
volume = {85:2}
journal type = {AC}
place = {Italy}
year = {1985}
month = {Feb}
}
title = {Small-amplitude limit of the spectral transform for the periodic Korteweg-de Vries equation}
author = {Osborne, A R, and Bergamasco, L}
abstractNote = {The inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered. In the limiting process we find that the widths of the forbidden bands approach the amplitudes of the Fourier spectrum. The number of spectral bands is estimated from Fourier theory and depends explicitly on the assumed spatial discretization in the wave amplitude function (potential). This allows one to estimate the number of degrees of freedom in a discrete (and, therefore, finite-banded) potential. An essential feature of the calculations is that all results for the periodic problem are cast in terms of the infinite-line reflection and transmission coefficients b(k), a(k). Thus the connection between the whole-line and periodic problems is clear at every stage of the computations.}
doi = {10.1007/BF02721563}
journal = []
volume = {85:2}
journal type = {AC}
place = {Italy}
year = {1985}
month = {Feb}
}