Non existence of L2?compact solutions of the Kadomtsev?Petviashvili II equation

Abstract

We prove that there is no nontrivial solution of the Kadomtsev–Petviashvili II equation (KP II equation) \({{ (u_t+u_{{xxx}}+uu_x)_x+u_{{yy}}=0,\quad (x,y)\in {{\bf{ R}}}^2, }}\) which is L 2 compact (i.e. uniformly localized in L 2 norm) and travel to the right in the x variable. This result extends the previous work of de Bouard and Saut [3] stating that there is no traveling wave solution for the KP II equation. The proof uses a monotonicity property of the L 2 mass for solutions of the KP II equation (similar to the one for the KdV equation [12], [14]) and two virial type relations. The result still holds for some natural generalizations of the KP II equation (general nonlinearity, higher dispersion) and does not rely on the integrability of the equation.