Symmetry and decay of traveling wave solutions to the Whitham equation
  This paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model. It is shown that all supercritical solitary wave solutions are symmetric and monotone on either side of the crest. The proof is based on a priori decay estimates and the method of moving planes. Furthermore, a close relation between symmetric and traveling wave solutions is established. 
Gabriele Bruell, Mats Ehrnstr\"om, and Long Pei
math.AP