We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ∂x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with peri...

This paper consider the Cauchy problem for Schr$ {\rm \ddot{o}} $dinger equation coupled with stochastic Benjamin-Ono equation. A priori estimates integral and nonlinear terms corresponding to coupling system are achieved by using Fourier transform restriction method introduced Bourgain. It is shown that locally well-posed as initial data in appropriate Sobolev spaces.

We consider the Benjamin–Ono equation on torus with an additional damping term smallest Fourier modes ( $$\cos $$ and $$\sin ). first prove global well-posedness of this in $$L^2_{r,0}(\mathbb {T})$$ . Then, we describe weak limit points trajectories when time goes to infinity, show that these are strong points. Finally, boundedness higher-order Sobolev norms for equation. Our key tool is Birkh...

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation ∂tu+ |∂x| ∂xu+ uux = 0, u(x, 0) = u0(x), is locally well-posed in the Sobolev spaces H for s > 1 − α if 0 ≤ α ≤ 1. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru [13] to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and T...

Considering the Cauchy problem for the modified finite-depthfluid equation ∂tu− Gδ(∂ 2 xu)∓ u 2ux = 0, u(0) = u0, where Gδf = −iF [coth(2πδξ)− 1 2πδξ ]Ff , δ&1, and u is a real-valued function, we show that it is uniformly globally well-posed if u0 ∈ Hs (s ≥ 1/2) with ‖u0‖L2 sufficiently small for all δ&1. Our result is sharp in the sense that the solution map fails to be C in Hs(s < 1/2). More...