Existence and singularities of solutions to an integrable equation governing short-waves in a long-wave model

This paper is concerned with the Cauchy problem of a new periodic integrable equation governing short-waves in a long-wave model. We first establish the local well-posedness for its derivative equation, which correspondingly implies the local existence and multivaluedness of solutions to the model. Then, we obtain a blow up scenario for these solutions. Finally, we show that the equation has strong solutions that develop singularities in finite time. (C) 2010 American Institute of Physics. [doi:10.1063/1.3488968]