Instabilities and oscillations of one- and two-dimensional Kadomtsev-Petviashvili waves and solitons II. Linear to nonlinear analysis
UNSPECIFIED. (2002) Instabilities and oscillations of one- and two-dimensional Kadomtsev-Petviashvili waves and solitons II. Linear to nonlinear analysis. PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 458 (2021). pp. 1231-1244. ISSN 1364-5021Full text not available from this repository.
We further investigate the dynamics of nonlinear structures that arise from the Kadomtsev-Petviashvili equation. When the analysis is linear, assuming perturbations grow exponentially in time, we find the growth rates of two important instabilities numerically. These are the only purely growing modes. The wavelength-doubling instability is seen to dominate its rival, that of Benjamin and Feir, at least when the amplitude of the wave is not too large. Approximate formulae, found to higher order than in part I (referenced in 1), are checked against the numerically found values. The models are seen to be better than expected. For the dominant wavelength-doubling instability, our model extends beyond the assumed region of validity. It is surprisingly close, almost up to the soliton limit.
When we depart from linear stability analysis and include terms nonlinear in the perturbation, a simple analysis shows that the linear instability eventually drives a doubly space-periodic hyperbolic secans pulse in time. After a long time, initial conditions are reproduced. A proof that the maximum amplitude achieved by the perturbation is approximately proportional to the linear growth rate is given within the limitations of the calculation. This fact was suspected from numerics. A second class of possible dynamic behaviour, not arising from initially linear growth of a perturbation, is found. This class involves fully two-dimensional stationary solutions and their possible oscillations.
|Item Type:||Journal Article|
|Journal or Publication Title:||PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES|
|Publisher:||ROYAL SOC LONDON|
|Official Date:||8 May 2002|
|Number of Pages:||14|
|Page Range:||pp. 1231-1244|
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