Blow up and instability of solitary wave solutions to a generalized Kadomtsev–Petviashvili equation and two-dimensional Benjamin–Ono equations

Abstract

Let with p being the ratio of an even to an odd integer. For the generalized Kadomtsev–Petviashvili equation, coupled with Benjamin–Ono equations, in the form it is proved that the solutions blow up in finite time even for those initial data with positive energy. As a by-product, it is proved that for all , the solitary waves are strongly unstable if . This result, even in a special case , improves a previous work by Liu (Liu 2001 Trans. AMS 353 , 191–208) where the instability of solitary waves was proved only in the case of .