Continuity and Analyticity for the Generalized Benjamin–Ono Equation

This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces Hs(R) with s>3/2. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in Hr-topology for all 0≤r<s with exponent α depending on s and r. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.

Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds

This paper is dedicated to proving the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by $$\mathcal {U}_N$$ U N . We construct generalized action–angle coordinates which establish a real analytic symplectomorphism from $$\mathcal {U}_N$$ U N onto some open convex subset of $${\mathbb {R}}^{2N}$$ R 2 N and allow to solve the equation by quadrature for any such initial datum. As a consequence, $$\mathcal {U}_N$$ U N is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard–Kappeler (Commun Pure Appl Math, 2020. 10.1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on $$\mathcal {U}_N$$ U N is given by a polynomial characterization and a spectral characterization of the manifold $$\mathcal {U}_N$$ U N . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup.

A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

A Stochastic Weakly Damped, Forced KdV-BO Equation

We investigate the long time behavior of the damped, forced KdV-BO equation driven by white noise. We first show that the global solution generates a random dynamical system. By energy type estimates and dispersive properties, we then prove that this system possesses a weak random attractor in the spaceH1(ℝ).