Ill-posedness estimation of the derivative schrödinger equation with a fourth-order power nonlinear term in H ^s

This article investigates a defocusing derivative nonlinear Schrödinger equation with a fourth-order power nonlinearity. A family of soliton solutions is constructed, and a strong instability phenomenon is proven for the associated flow in Sobolev spaces (

H^s

). Using a refined fourier integral estimate,

(1/4)

is identified as an upper threshold for the instability index in

H^s

.