We consider a higher-dimensional version of the Benjamin-Ono (HBO) equation in 2D setting: , which is -critical, and investigate properties solutions both analytically numerically. For generalized (fractional gKdV) after deriving Pohozaev identities, we obtain nonexistence conditions for solitary wave solutions, then prove uniform bounds energy space or conditional global existence, radiation r...

In a recent work [12], Ionescu and Kenig proved that the Cauchy problem associated to the Benjamin-Ono equation is well-posed in L(R). In this paper we give a simpler proof of Ionescu and Kenig’s result, which moreover provides stronger uniqueness results. In particular, we prove unconditional well-posedness in Hs(R), for s > 1 4 .

Explicit counterexamples to bilinear estimates related to the Benjamin-Ono equation in the periodic setting are calculated for functions of zero mean value. As a consequence, certain bilinear estimates fail to hold in spite of the analyticity of the flow map. The latter has been shown recently by L. Molinet.

We prove that the Benjamin–Ono initial-value problem is locally well-posed for small data in the Banach spaces H̃σ(R), σ ≥ 0, of complex-valued Sobolev functions with special low-frequency structure.

This work studies the rotation-generalized Benjamin-Ono equation which is derived from the theory of weakly nonlinear long surface and internal waves in deep water under the presence of rotation. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

We present a spectrally accurate numerical method for finding non-trivial timeperiodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the in...

A hierarchy of pairwise commuting Hamiltonians for the quantum periodic Benjamin–Ono equation is constructed by using the Lax matrix. The eigenvectors of these Hamiltonians are Jack symmetric functions of infinitely many variables x1, x2, . . .. This construction provides explicit expressions for the Hamiltonians in terms of the power sum symmetric functions pn = x n 1 + x n 2 + · · · and is ba...

We show that the Benjamin-Ono equation is globally well-posed in H s (R) for s ≥ 1. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in H s for any s [15]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.

We prove that the complex-valued modified Benjamin-Ono (mBO) equation is locally wellposed if the initial data φ belongs to Hs for s ≥ 1/2 with ‖φ‖L2 sufficiently small without performing a gauge transformation. Hence the real-valued mBO equation is globally wellposed for those initial datas, which is contained in the results of C. Kenig and H. Takaoka [25] where the smallness condition is not ...

We present a spectrally accurate numerical method for finding non-trivial timeperiodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the in...