Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

Abstract

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schrödinger equation with the derivative quartic nonlinearity ${\partial }_x({\overline{u}}^4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d\ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla |(u^m)$ with $(m-1)d\ge 4$.

References in corpus (2)

• Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces
• Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity

Cited by in corpus (3)

• Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities
• Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity
• Well-posedness of generalized KdV and one-dimensional fourth-order derivative nonlinear Schrödinger equations for data with an infinite $L^2$ norm