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| A generalised solitary wave is a travelling wave consisting of a localised central part and periodic non-decaying oscillations extending to infinity. They arise in weakly nonlinear equations due to a resonance between a long wave and a short wave with finite wave number. An important example is the water wave problem with weak surface tension (Bond number less than 1/3). The localised part is then described to leading order by the KdV equation. Although the KdV soliton is stable, instabilities may arise from the periodic wave trains at infinity. I will discuss this question for a simpler model equation. The Kawahara equation is a fifth order version of the KdV equation, modelling waves with Bond number close to 1/3. It's known to have generalised solitary-wave solutions. There is also a 2D version called the 5th order KP equation. I will discuss the spectral instability for the generalised solitary waves (seen as generalised line solitary waves of the 5th order KP equation) to perturbations in the transverse direction. |
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