## Abstract

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation i∂_{t}u + ∂_{x}^{2}u + i|u|^{2σ}∂_{x}u = 0. The equation has a two-parameter family of solitary wave solutions of the form ϕ_{ω,c}(x) = φ_{ω,c}(x) exp { i _{2}^{c} x − _{2σ}^{i}_{+ 2}^{Z}_{−∞}^{x} φ^{2}_{ω,c}^{σ} (y)dy }. Here φ_{ω,c} is some real-valued function. It was proved in [29] that the solitary wave solutions are stable if −2√ω < c < 2z_{0}^{√}ω, and unstable if 2z_{0}^{√}ω < c < 2√ω for some z_{0} ∈ (0, 1). We prove the instability at the borderline case c = 2z_{0}^{√}ω for 1 < σ < 2, improving the previous results in [7] where 7/6 < σ < 2.

Original language | English |
---|---|

Pages (from-to) | 339-375 |

Number of pages | 37 |

Journal | Mathematical Research Letters |

Volume | 27 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2020 |