Multidimensional shallow water waves

Mark J. Ablowitz Department of Applied Mathematics University of Colorado

Hopeman 224

Multidimensional interacting “X,Y” type soliton waves and ones with more complex “web” structure can frequently be seen in shallow water on flat beaches. These waves are related to solutions of the Kadomtsev-Petviashvili (KP) equation which is a two-dimensional extension of the one-dimensional Korteweg-deVries (KdV) equation. Another application to the KP equation is dispersive shock waves (DSWs). The study of DSWs in the KP equation with initial data varying across a parabolic front can be reduced exactly to a one-dimensional cylindrical KdV (cKdV) equation.  Comparing asymptotic analysis and numerics for the cKdV and the KP equation shows excellent agreement. Similar DSW type structures have been observed in shallow water bore tides near Anchorage, Alaska.

*This lecture is in honor of my former undergraduate Professor/mentor: Alfred Clark whose course: `MAS 201' (now replaced by ME 201), I took in the fall 1965 — 50 years ago. Interestingly, one of  Al Clark’s earliest papers was on shallow water waves.

**After the interacting `X,Y'  soliton wave research was published (Phys. Rev. 2012) the article was highlighted in Physics Today and subsequently was the subject of numerous news articles.