An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type

Jared C. Bronski; Mathew A. Johnson; Todd Kapitula

doi:10.1017/S0308210510001216

An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type

Jared C. Bronski, Mathew A. Johnson, Todd Kapitula

Mathematics

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Abstract

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg-de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L ²-critical KdV-4 equation that arises in the study of blow-up and the KdV-1/2 equation, which is an idealized model for plasmas.

Original language	English (US)
Pages (from-to)	1141-1173
Number of pages	33
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume	141
Issue number	6
DOIs	https://doi.org/10.1017/S0308210510001216
State	Published - Dec 2011

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General Mathematics

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title = "An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type",

abstract = "We consider the stability of periodic travelling-wave solutions to a generalized Korteweg-de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L 2-critical KdV-4 equation that arises in the study of blow-up and the KdV-1/2 equation, which is an idealized model for plasmas.",

author = "Bronski, {Jared C.} and Johnson, {Mathew A.} and Todd Kapitula",

note = "Funding Information: J.C.B. gratefully acknowledges support from the National Science Foundation under NSF Grant no. DMS-0807584. M.A.J. gratefully acknowledges support from a National Science Foundation Postdoctoral Fellowship under NSF Grant DMS-0902192. Funding Information: T.K. gratefully acknowledges the support of a Calvin Research Fellowship and the National Science Foundation under Grant DMS-0806636.",

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doi = "10.1017/S0308210510001216",

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AU - Kapitula, Todd

N1 - Funding Information: J.C.B. gratefully acknowledges support from the National Science Foundation under NSF Grant no. DMS-0807584. M.A.J. gratefully acknowledges support from a National Science Foundation Postdoctoral Fellowship under NSF Grant DMS-0902192. Funding Information: T.K. gratefully acknowledges the support of a Calvin Research Fellowship and the National Science Foundation under Grant DMS-0806636.

PY - 2011/12

Y1 - 2011/12

N2 - We consider the stability of periodic travelling-wave solutions to a generalized Korteweg-de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L 2-critical KdV-4 equation that arises in the study of blow-up and the KdV-1/2 equation, which is an idealized model for plasmas.

AB - We consider the stability of periodic travelling-wave solutions to a generalized Korteweg-de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L 2-critical KdV-4 equation that arises in the study of blow-up and the KdV-1/2 equation, which is an idealized model for plasmas.

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An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type

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