We prove the invariance of the Gibbs measure for the periodic SchrödingerBenjamin-Ono system (when the coupling parameter |γ| 6= 0, 1) by establishing a new local well-posedness in a modified Sobolev space and constructing the Gibbs measure (which is in the sub-L setting for the Benjamin-Ono part.) We also show the ill-posedness result in H(T)×H 1 2 (T) for s < 1 2 when |γ| 6= 0, 1 and for any ...

We classify all bifurcations from traveling waves to non-trivial time-periodic solutions of the Benjamin-Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of non-trivial solutions beyond the realm of linear theory. These paths are found to either re-connect with a different traveling wave or to blow up. In the lat...

We consider the zero-dispersion limit for Benjamin–Ono equation on torus. prove that when initial data is bell shaped, exists in weak sense and uniform every compact time interval. Moreover, equal to signed sum of branches multivalued solution inviscid Burgers obtained by method characteristics. This result similar one Miller Xu real line decaying positive data. also establish some precise asym...

In this paper we study the initial-value problem associated with dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation,ut+Dxa+1∂xu+uxyy+uux=0,a∈(0,1). More specifically, persistence property of solution in weighted anisotropic Sobolev spacesH(1+a)s,2s(R2)∩L2((x2r1+y2r2)dxdy), for appropriate s, r1 and r2. By establishing unique continuation properties also show that our results are sh...

The objective of this paper is two-fold: firstly, we develop a local and global (in time) wellposedness theory of a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schrödinger-Benjamin-Ono system) for lowregularity initial data in both periodic and continuous cases; secondly, a family of new periodic travelling wa...

The objective of this paper is two-fold: firstly, we develop a local and global (in time) wellposedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schrödinger-Benjamin-Ono system) for lowregularity initial data in both periodic and continuous cases; secondly, a family of new periodic traveling wa...