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Existence and qualitative theory of stratified solitary water waves Walsh, Samuel


In this talk, we will report some recent results concerning two-dimensional gravity solitary water waves with hereogeneous density. The fluid domain is assumed be bounded below by an impenetrable flat ocean bed, while the interface between the water and vacuum above is a free boundary. Our main existence result states that, for any smooth choice of upstream velocity and streamline density function, there exists a path connected set of such solutions that includes large-amplitude surface waves. Indeed, this solution set can be continued up to (but does not include) an ``extreme wave`` that possess a stagnation point. We will also discuss a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the Froude number from above and below that are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores for stratified surface waves; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry. This is joint work with R. M. Chen and M. H. Wheeler.

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