Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

The Benjamin-Ono hierarchy with asymptotically reflectionless initial data in the zero-dispersion limit

Pages: 117 – 130



Peter D. Miller (Department of Mathematics, University of Michigan, Ann Arbor)

Zhengjie Xu (Department of Mathematics, University of Michigan, Ann Arbor)


We study the Benjamin-Ono hierarchy with positive initial data of a general type, in the limit when the dispersion parameter tends to zero. We establish simple formulae for the limits (in appropriate weak or distributional senses) of an infnite family of simultaneously conserved densities in terms of alternating sums of branches of solutions of the inviscid Burgers hierarchy.


internal waves, Benjamin-Ono equation, integrable hierarchies, dispersionless asymptotics

2010 Mathematics Subject Classification

35Q35, 35Q53, 37K10, 37K40

Published 14 October 2011