On decay of the solutions for the dispersion generalized Benjamin--Ono and Benjamin--Ono equations

Abstract

We show that uniqueness results of the kind obtained for KdV and Schrödinger equations ([7], [28]), are not valid for the dispersion generalized--Benjamin--Ono equation in the weighted Sobolev spaces $$ H^s(\mathbb R)\cap L^2(x^{2r}dx), $$ for appropriated $s$ and $r$. In particular, we obtain that the uniqueness result proved for the dispersion generalized--Benjamin--Ono equation ([13]), is not true for all pairs of solutions $u_1\neq 0$ and $u_2\neq 0$. To achieve these results, we employ the techniques present in our recent work [6]. We also improve some Theorems established for the dispersion generalized--Benjamin--Ono equation and for the Benjamin--Ono equation ([13], [12]).