Dispersion estimates for fourth order Schrödinger equations

Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces

“…The 4NLS with different nonlinearities was studied by several authors, one can see [1,6,15,16,17,22,23,33,34,30] and references therein. In [33] and [16], by using the method of Fourier restriction norm, Segata, Huo and Jia obtained that (1.4) is local well-posed in H s (R) (s 1/2).…”

“…The case |k (2) 1 | = max(|k (1) 1 |, |k (2) 1 |) can be handled in an analogous way as above. Hence, we obtain the result, as desired.…”

Singular solutions of theL2-supercritical biharmonic nonlinear Schrödinger equation

“…This is analogous to the NLS, where the critical case is σd = 2. BNLS solutions preserve the power (L 2 norm) P (t) ≡ P (0), P = ||ψ|| In [BAKS00], Ben-Artzi, Koch and Saut proved that when σ is in the H 2 -subcritical regime…”

On the focusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space

“…The proof is based on the scaling technique instead of using a dedicate dispersive estimate of [1] for the fundamental solution of the homogeneous fourth-order Schrödinger equation. Note that the estimate (2.3) is exactly the one given in [27], [28] or [29] where the author considered (p, q) and (a, b) are either sharp Schrödinger admissible, i.e.…”