Local and global well-posedness for some dispersive equations in modulation spaces

Abstract

We consider the initial value problem (IVP) associated with a system consisting of modified Korteweg-de Vries (mKdV) type equations ∂tv + ∂3 xv + ∂x(vw2) = 0, v(x, 0) = v0(x), ∂tw + α∂3 xw + ∂x(v2w) = 0, w(x, 0) = w0(x), (3) using the theory of T. Oh and Y. Wang [31] we derive trilinear estimates and use it in the contraction argument, so we prove local well-posedness (LWP) to the IVP (3) for given data in the modulation spaces M2,p s (R) ×M2,p s (R), with s ≥ 1 4 , 2 ≤ p < ∞ and α ̸= 0. Also, at the second part of this work we consider the initial value problem (IVP) associated to the higher order nonlinear Schrödinger (h-NLS) equation with variable coefficients ∂tu + ia(t)∂2 xu + b(t)∂3 xu + ic(t)|u|2u + c1(t)|u|2∂xu = 0, u(x, 0) = u0(x), (4) using the theory of Killip, Visan, Zhang [24], Oh, Wang [30], [31], Carvajal, Gamboa and Santos [8], so we establish a priori estimates and prove global well-posedness (GWP) to the IVP (4) for given data in the modulation spaces M2,p s (R), with s ≥ 1 4 and 2 ≤ p < ∞.