Abstract
We utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem:
when ∈ → 0. We refine the Lax-Levermore theory, which gives the weak limit of the solution as ∈ → 0, by introducing a quantum condition. This allows us to calculate the waveform of the local oscillations up to phase-shifts when ∈ is small.
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References
Buslaev, V. S., and Fomin, V. N., (1962) ‘An inverse scattering problem for the one-dimensional Schrödinger equation on the entire axis’, (in Russian), Vestnik Leningrad Uniiv. 17, 56–64.
Cohen, A., and Kappeler, T., (1985) ‘Scattering and inverse scattering for steplike potentials in the Schrödinger equation’, Indiana U. Math. Jour. 34, 127–180.
Flaschka, H., Forest, M.G., and McLaughlin, D. W., (1980) ‘Multiphase averaging and the inverse spectral solutions of the Korteweg—de Vries equation’, Comm. Pure Appl. Math. 33, 739–784.
Lax, P.D., and Levermore, C.D., (1983) ‘The small dispersion limit of the Korteweg-de Vries equation I, II, III’, Comm. Pure Appl. Math. 36, 253–290, 571–593, 809––829.
Levermore, C. D., (1988) ‘The hyperbolic nature of the zero dispersion KdV limit’, Comm. P.D.E., 495–614.
Venakides, S., (1985) ‘The zero dispersion limit of the Korteweg-de Vries equation with nontrivial reflection coefficient’, Comm. Pure Appl. Math. 38, 125–155.
Venakides, S., (1985) ‘The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation’, Comm. Pure Appl. Math. 38, 883–909.
Venakides, S., (1987) ‘The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data’, AMS Trans. 301, 189–225.
Venakides, S., (1989) ‘The continuum limit of theta functions’, Comm. PUre Appl. Math., 42, 711–728.
Bender C.M., and Orszag, Steven A., (1978) Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Publishers.
Venakides, S., ’The Korteweg—de Vries equation with small dispersion: higher order Lax—Levermore theory, Comm. Pure Appl. Math., in press
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© 1991 Kluwer Academic Publishers
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Venakides, S. (1991). The Korteweg-de Vries Equation with Small Dispersion: Higher Order Lax-Levermore Theory. In: Spigler, R. (eds) Applied and Industrial Mathematics. Mathematics and Its Applications, vol 56. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1908-2_19
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DOI: https://doi.org/10.1007/978-94-009-1908-2_19
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7351-6
Online ISBN: 978-94-009-1908-2
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