Solitary Waves of the Regularized Short Pulse and Ostrovsky Equations

We derive a model for the propagation of short pulses in nonlinear media. The model is a higher-order regularization of the short-pulse equation (SPE). The regularization term arises as the next term in the expansion of the susceptibility in derivation of the SPE. Without the regularization term the...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 2009-01, Vol.41 (5), p.2088-2106
Main Authors: Costanzino, Nicola, Manukian, Vahagn, Jones, Christopher K. R. T.
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Derivation
Discontinuity
Fourier transforms
Mathematical analysis
Mathematical models
Nonlinearity
Propagation
Regularization
Short pulses
Solitary waves
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Summary:We derive a model for the propagation of short pulses in nonlinear media. The model is a higher-order regularization of the short-pulse equation (SPE). The regularization term arises as the next term in the expansion of the susceptibility in derivation of the SPE. Without the regularization term there do not exist traveling pulses in the class of piecewise smooth functions with one discontinuity. However, when the regularization term is added, we show, for a particular parameter regime, that the equation supports smooth traveling waves which have structure similar to solitary waves of the modified Korteweg-deVries equation. The existence of such traveling pulses is proved via the Fenichel theory for singularly perturbed systems and a Melnikov-type transversality calculation. Corresponding statements for the Ostrovsky equations are also included.
ISSN:0036-1410
1095-7154
DOI:10.1137/080734327