We consider a higher dimensional version of the Benjamin-Ono equation, ∂tu−R 1 ∆u+u∂x 1 u = 0, where R 1 denotes the Riesz transform with respect to the first coordinate. We first establish sharp space-time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in L 2 -Sobolev spaces H s (R d ), with s > 5/3 if d = 2 and s > d/2 + 1/2 if d 3. We also provide ill-posedness results.With d = 1, the available local well-posedness theory has been based on compactness methods. Indeed, Molinet, Saut and Tzvetkov [19] proved that the problem cannot be solved in L 2 -Sobolev spaces H s by Picard iteration. We will show that

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Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg–de Vries equation, which comprises the Burgers–Hilbert equation and dispersive effects weaker than those of the Benjamin–Ono equation.

We consider a higher‐dimensional version of the Benjamin‐Ono (HBO) equation in the 2D setting: ut−R1normalΔu+12false(u2false)x=0,(x,y)∈R2$u_t- \mathcal {R}_1 \Delta u + \frac{1}{2}(u^2)_x=0, (x,y) \in \mathbb {R}^2$, which is L2$L^2$‐critical, and investigate properties of solutions both analytically and numerically. For a generalized equation (fractional 2D gKdV) after deriving the Pohozaev identities, we obtain nonexistence 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$x$‐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 versus 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$x$‐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.