Well-posedness and ill-posedness for the fourth order cubic nonlinear Schrödinger equation in negative Sobolev spaces

Propagation of radius of analyticity for solutions to a fourth order nonlinear Schr\”odinger equation

“…and the standard transference principle. To be more precise ( 28) is proved in [28] with D replaced by |D|. However, the proof can be easily modified to deduce that (28) still holds.…”

“…are conserved by the flow of (4NLS). Low regularity well-posedness for (4NLS) was recently studied by Seong [28]. The author proved that (4NLS) is locally and globally well-posed for initial data f ∈ H s (R) , s −1/2.…”

“…As a consequence of the embedding H σ 0 ,2 (R) → H 2 (R) and the existing well-posedness theory in H 2 (R) (see[28]), the Cauchy problem (4NLS) has a unique, smooth solution for all time, given initial data f ∈ H σ 0 ,2 .…”

Well‐posedness of generalized KdV and one‐dimensional fourth‐order derivative nonlinear Schrödinger equations for data with an infinite L2 norm

“…For equations without derivative on nonlinear term, the well‐posedness theory in $$H^{s}$$ has been studied in Refs. 13–16. Hayashi et al 17–19 studied the well‐posedness and scattering theory of (4NLS) in weighted Sobolev spaces.…”

Ill-posedness of quintic fourth order Schrödinger equation

“…Moreover, Kishimoto [20] showed that the norm inflation phenomenon also occur for the higher dimensional Schrödigner equations spaces of low regularity. See also the work by Choffrut and Pocuvnicu [4] for the cubic fractional Schrödinger equations, that by Seong [30] for the cubic fourth order Schrödinger equation, and that by Dinh [9] for the fourth order Schrödinger equations with more general nonlinearities.…”