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85037226166
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The Schwarz inequality implies (Formula presented) and here for the symmetrical case (Formula presented)
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The Schwarz inequality implies (Formula presented) and here for the symmetrical case (Formula presented)
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23
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85037203301
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Unstable solutions are numerically obtained with the AUTO94 computer code from E. J. Doedel, Concordia University, Montreal
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Unstable solutions are numerically obtained with the AUTO94 computer code from E. J. Doedel, Concordia University, Montreal.
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24
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85037218786
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From the definition of the Gauss-Hermite modes, it can be qualitatively understood that (Formula presented) (Formula presented) since higher modes have higher spatial extension and therefore lower average spatial values; the parameter (Formula presented) is the spatial integral of the mode to the fourth order and tends to zero for increasing order of the mode. From the same arguments one has (Formula presented) (Formula presented). These qualitative arguments have been checked by explicit numerical integration
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From the definition of the Gauss-Hermite modes, it can be qualitatively understood that (Formula presented) (Formula presented) since higher modes have higher spatial extension and therefore lower average spatial values; the parameter (Formula presented) is the spatial integral of the mode to the fourth order and tends to zero for increasing order of the mode. From the same arguments one has (Formula presented) (Formula presented). These qualitative arguments have been checked by explicit numerical integration.
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26
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85037194567
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F. Prati, Ph.D. dissertation, University of Zurich, 1992 (unpublished)
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F. Prati, Ph.D. dissertation, University of Zurich, 1992 (unpublished).
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27
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85037244192
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For the (Formula presented)-(Formula presented) case, for instance, one has (Formula presented), (Formula presented), and (Formula presented). Equation (4.20) leads to (Formula presented)
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For the (Formula presented)-(Formula presented) case, for instance, one has (Formula presented), (Formula presented), and (Formula presented). Equation (4.20) leads to (Formula presented).
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