The dual modified Korteweg-de Vries–Fokas–Qiao equation: Geometry and local analysis

Abstract

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 equation, and also known to arise as the dual of the modified Korteweg-de Vries equation thanks to work by Fokas, Fuchssteiner, Olver, and Rosenau. We present a quadratic pseudo-potential, we compute infinite sequences of local and nonlocal conservation laws, and we construct an infinite-dimensional Lie algebra of symmetries which contains a semi-direct sum of the $sl(2,\mathbb {R})$sl(2,R)-loop algebra and the centerless Virasoro algebra. As an application we prove a theorem on the existence of smooth solutions, and we construct some explicit examples. Moreover, we consider the Cauchy problem and we prove existence and uniqueness of weak solutions in the Sobolev space $H^{q+2}({\mathbb R})$Hq+2(R), q > 1/2.