Ill-posedness of an NLS-type equation with derivative nonlinearity on the torus
A talk in the Oberseminar Analysis series by
Nobu Kishimoto from RIMS, Kyoto
| Abstract: | We consider nonlinear Schrödinger equations with third-order dispersion and derivative nonlinearity on the one-dimensional torus:
$$ i\partial _tu + \partial _x^2u + i\partial _x^3u = c_1 |u|^2u +ic_2 \partial _x(|u|^2u) + i\gamma \partial _x(|u|^2)u, \qquad t\in \mathbb{R},\quad x\in \mathbb{R}/2\pi \mathbb{Z} ,$$
where $c_1,c_2$ are real constants and $\gamma$ is a complex constant. This equation is regarded as a mathematical model for the photonic crystal fiber oscillator, and the last term, with the coefficient $\gamma$ having non-zero imaginary part, is related to the intrapulse Raman scattering effect, which is not negligible for ultrashort optical pulses. Without the Raman scattering term (i.e., $\Im \gamma =0$), or for the non-periodic problem with any complex $\gamma$, the associated Cauchy problem is known to be locally well-posed in Sobolev spaces. We show that in the periodic setting the Raman scattering term causes ill-posedness (more precisely, non-existence of local-in-time solutions) of the Cauchy problem in Sobolev and Gevrey spaces. We also mention that the Cauchy problem is uniquely solvable in the analytic function space. This talk is based on a jount work with Yoshio Tsutsumi (Kyoto University).
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