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On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation. (English) Zbl 0962.35505

Summary: One-dimensional solitary wave solutions of the Kadomtsev-Petviashvili equation were shown to be unstable to long-wavelength transverse disturbances by Kadomtsev and Petviashvili, in the positive dispersion case. Here we show that there is a short-wavelength cutoff for the instability, which is associated with a bifurcation to transversely modulated solitary waves, and we identify the dominant mode of instability, by finding explicitly all the exponentially unstable modes of the linearized equation for perturbations of the solitary wave. No unstable modes are found in the negative dispersion case.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B35 Stability in context of PDEs
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