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On the Singular Behavior of Spatially Dependent Nonlinear Wave Envelopes

Richard Haberman
SIAM Journal on Applied Mathematics
Vol. 32, No. 1 (Jan., 1977), pp. 154-163
Stable URL: http://www.jstor.org/stable/2100290
Page Count: 10
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On the Singular Behavior of Spatially Dependent Nonlinear Wave Envelopes
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Abstract

The evolution of a small amplitude spatially dependent wave envelope is analyzed in the case in which the envelope is linearly unstable. The energy first tends to focus at one point in space. Once the wave envelope becomes sufficiently sharply peaked, a singularity may or may not develop depending only on the nonlinear, diffusive, and dispersive parameters. In certain cases, the structure, position, and time of the first singularity is determined and shown to be related to the initial conditions by the method of matched asymptotic expansions.

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