Mathematical Research Letters

Volume 25 (2018)

Number 5

Ill-posedness of the third order NLS equation with Raman scattering term

Pages: 1447 – 1484

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n5.a5

Authors

Nobu Kishimoto (Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, Japan)

Yoshio Tsutsumi (Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, Japan)

Abstract

We consider the ill-posedness and well-posedness of the Cauchy problem for the third order NLS equation with Raman scattering term on the one dimensional torus. It is regarded as a mathematical model for the photonic crystal fiber oscillator. Regarding the illposedness, we show the nonexistence of solutions in the Sobolev space and the norm inflation of the data-solution map at the origin under slightly different conditions, respectively. We also prove the local unique existence of solutions in the analytic function space.

The first author N.K is partially supported by JSPS KAKENHI Grant-in-Aid for Young Researchers (B) (16K17626).

The second author Y.T is partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) (17H02853) and Grant-in-Aid for Exploratory Research (16K13770).

Received 8 June 2017

Published 1 February 2019