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

TL;DR

This paper investigates the decay and uniqueness properties of solutions to the dispersion generalized-Benjamin-Ono and Benjamin-Ono equations, showing that certain uniqueness results do not hold in specific weighted Sobolev spaces and improving existing theorems.

Contribution

The authors demonstrate the failure of some uniqueness results in weighted Sobolev spaces for these equations and improve previous theorems related to their decay properties.

Findings

Uniqueness results for KdV and Schrödinger do not extend to the dispersion generalized-Benjamin-Ono equation.

Certain pairs of solutions are shown to violate previously established uniqueness in specific spaces.

The paper refines existing theorems regarding decay and uniqueness for these equations.

Abstract

We show that uniqueness results of the kind those obtained for KdV and Schr\"odinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spaces $$H^s(\R)\cap L^2(x^{2r}dx),$$ 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]).