Steady periodic flow induced by the Korteweg-de Vries equation

The Korteweg-de Vries equation can be derived in the hydrodynamical setting as an approximation to the full governing equations. The periodic solutions of the equation are expressed in terms of the Jacobian elliptic functions, and they describe a steady periodic surface wave profile. However, at the level of approximation which generates the Korteweg-de Vries equation, the velocity potential does not provide a suitable description of the flow. We propose a remedy for this situation by constructing a velocity potential which is compatible with the Korteweg-de Vries regime and describes a nontrivial fluid flow. We discuss some qualitative aspects of the flow.