Stability of lumps and wave collapse in water waves

来源 :International Conference Nonlinear Waves-Theory and Applicat | 被引量 : 0次 | 上传用户:hellojie
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  This study is concerned with the class of gravity-capillary lumps recently found on water of finite or infinite depth for Bond number, B < 1/3.In the near-linear limit, these lumps resemble locally confined wavepackets and can be approximated in terms of a par ticular steady solution (ground state) of an elliptic system of the Benney-Roskes-DaveyStewartson (BRDS) type.According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time owing to nonlinear focusing; the ground state, in fact, being on the borderline between existence and wave collapse sug gests that the newly discovered lumps are unstable.Based on the fifth-order KP equation, a model for gravity-capillary waves when B is close to 1/3, it is pointed out that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable (stable) below (above) this critical value.As a result, a small-amplitude lump, that is linearly unstable and would be prone to wave collapse according to the BRDS equations, depending on the perturbation, either decays into dispersive waves or evolves to an oscillatory state near a finite-amplitude stable lump.The effect of external forcing on the dynamics of lumps will also be discussed.
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