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Register a new userOn the integrability of the Benjamin-Ono equation on the torus
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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 which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy.
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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 $n$-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As $nrightarrowinfty$, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.
For initial data in Sobolev spaces $H^s(mathbb T)$, $frac 12 < s leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-frac 12) + epsilon}$, $0<epsilon ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.
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 and numerically. For a generalized equation (fractional 2D gKdV) after deriving the Pohozaev identities, we obtain non-existence conditions for solitary wave solutions, then prove uniform bounds in the energy space or conditional global existence, and investigate the radiation region, a specific wedge in the negative $x$-direction. We then introduce our numerical approach in a general context, and apply it to obtain the ground state solution in the 2D critical HBO equation, then show that its mass is a threshold for global vs. finite time existing solutions, which is typical in the focusing (mass-)critical dispersive equations. We also observe that globally existing solutions tend to disperse completely into the radiation in this nonlocal equation. The blow-up solutions travel in the positive $x$-direction with the rescaled ground state profile while also radiating dispersive oscillations into the radiative wedge. We conclude with examples of different interactions of two solitary wave solutions, including weak and strong interactions.
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.
In this work we continue our study initiated in cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in cite{GFGP} do not extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in cite{GFGP} under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.
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