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The effect of weak shear on finite-amplitude internal solitary waves. (English) Zbl 0978.76015

A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg-de Vries equation. The type of waves that occur is found to depend on the number and position of inflection points of the representation of shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores), which can have up to three characteristic length scales. The stability of these waves is generally found to decrease as the complexity of waves increases. These solutions suggest that kinks and solitary waves with multiple length scales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered, and again the solution is found to depend on the inflection points of amplitude representation of shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rank-ordered solitary waves, analogous to those generated in the framework of Korteweg-de Vries equation, and the second where two or more kinks connect regions of constant amplitude.

MSC:

76B25 Solitary waves for incompressible inviscid fluids
76B55 Internal waves for incompressible inviscid fluids
35Q51 Soliton equations
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