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.

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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

Accepted 22 March 2018

Published 1 February 2019