Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation

Well-posedness for a Higher-order Benjamin-ono Equation

In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation ∂tv − bH∂ xv + a∂ xv = cv∂xv − d∂x(vH∂xv + H(v∂xv)), where x, t ∈ R, v is a real-valued function, H is the Hilbert transform, a ∈ R, b, c and d are positive constants, is locally well-posed for initial data v(0) = v0 ∈ H(R), s ≥ 2 or v0 ∈ H(R) ∩ L(R; xdx), k ∈ Z+, k ≥ 2.

Perturbation theory for the Benjamin–Ono equation

We develop a perturbation theory for the Benjamin–Ono (BO) equation. This perturbation theory is based on the inverse scattering transform for the BO equation, which was originally developed by Fokas and Ablowitz and recently refined by Kaup and Matsuno. We find the expressions for the variations of the scattering data with respect to the potential, as well as the dual expression for the variat...

Multiple Soliton Solutions of Second-order Benjamin-ono Equation

We employ the idea of Hirota’s bilinear method, to obtain some new exact soliton solutions for high nonlinear form of Multiple soliton solutions of secondorder Benjamin-Ono equation. Multiple singular soliton solutions were obtained by this method. Moreover, multiple singular soliton solutions were also derived.

Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation (0.1) ∂tv − bH∂ 2 x v − aǫ∂ x v = cv∂xv − dǫ∂x(vH∂xv +H(v∂xv)), is globally well-posed in the energy space H(R). Moreover, we study the limit behavior when the small positive parameter ǫ tends to zero and show that, under a condition on the coefficients a, b, c and d, the solution vǫ t...

Benjamin-Ono Equation on a Half-Line