The sharp range of Sobolev spaces is determined in which the Cauchy problem for classical Zakharov system well-posed, includes existence solutions, uniqueness, persistence initial regularity, and real-analytic dependence on data. In addition, under a condition data Schr\"odinger equation at lowest admissible global well-posedness scattering proved. results cover energy-critical energy-supercrit...

We consider the general two-dimensional Zakharov-Shabat systems, which appear in application of inverse scattering transform (IST) to an important class nonlinear partial differential equations (PDEs) called integrable systems. study their integrability meaning Galois theory, i.e., solvability by quadrature. It becomes a key for obtaining analytical expressions solutions PDEs using IST. For wid...

In this article we study the smoothness properties of solutions to a two-dimensional coupled Zakharov-Kuznetsov system. We show that equations dispersive nature leads gain in regularity for solution. particular, if initial data (u0,v0) possesses certain and sufficient decay as x → ∞, then solution (u(t),v(t)) will be smoother than (u0, v0) 0 < t ≤ T where is existence time