Optimal small data scattering for the generalized derivative nonlinear Schrödinger equations
Peer reviewed, Journal article
Accepted version
Åpne
Permanent lenke
https://hdl.handle.net/11250/2731961Utgivelsesdato
2020Metadata
Vis full innførselSamlinger
- Institutt for matematiske fag [2533]
- Publikasjoner fra CRIStin - NTNU [38576]
Originalversjon
Journal of Differential Equations. 2020, 269 (9), 6422-6447. https://doi.org/10.1016/j.jde.2020.05.001Sammendrag
In this work, we consider the following generalized derivative nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\partial_{xx} u +i |u|^{2\sigma}\partial_x u=0, \quad (t,x)\in \R\times \R. \end{align*} We prove that when $\sigma\ge 2$, the solution is global and scattering when the initial data is small in $H^s(\R)$, $\frac 12\leq s\leq1$. Moreover, we show that when $0<\sigma<2$, there exist a class of solitary wave solutions $\{\phi_c\}$ satisfying $$ \|\phi_c\|_{H^1(\R)}\to 0, $$ when $c$ tends to some endpoint, which is against the small data scattering statement. Therefore, in this model, the exponent $\sigma\ge2$ is optimal for small data scattering. We remark that this exponent is larger than the short range exponent and the Strauss exponent.