The Dual Modified Korteweg-De Vries-Fokas-Qiao Equation: Geometry and Local Analysis

We study a bi-hamiltonian equation with cubic nonlinearity shown to appear in the theory of water waves by Fokas, derived by Qiao using the two-dimensional Euler equationalso known to arise as the dual of the modified Korteweg-de Vries equation thanks to work by Fokas, Fuchssteiner, OlverRosenau. We present a quadratic pseudo-potential, we compute infinite sequences of local and nonlocal conservation lawswe construct an infinite-dimensional Lie algebra of symmetries which contains a semi-direct sum of the sl(2,R)-loop algebra and the centerless Virasoro algebra. As an application we prove a theorem on the existence of smooth solutionswe construct some explicit examples. Moreover, we consider the Cauchy problem and we prove existence and uniqueness of weak solutions in the Sobolev space H q+2R,q > 1/2. © 2012 American Institute of Physics.