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We prove that if $u_1,,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)inR times [0,T]$ which agree in an open set $Omegasubset R times [0,T]$, then $u_1equiv u_2$. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.
We consider the generalized Benjamin-Ono (gBO) equation on the real line, $ u_t + partial_x (-mathcal H u_{x} + tfrac1{m} u^m) = 0, x in mathbb R, m = 2,3,4,5$, and perform numerical study of its solutions. We first compute the ground state solution
The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of $n$-solito
We consider a higher-dimensional version of the Benjamin-Ono (HBO) equation in the 2D setting: $u_t- mathcal{R}_1 Delta u + frac{1}{2}(u^2)_x=0, (x,y) in mathbb{R}^2$, which is $L^2$-critical, and investigate properties of solutions both analytically
In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(T)$. These are coordinates wh
We prove the discontinuity for the weak $ L^2(T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(T) $ as soon as $ s<0 $ and thus completes exactly the well-posedness result obtained by the author.