Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schrödinger equation in the critical frequency case

Abstract

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equationThe equation has a two-parameter family of solitary wave solutions of the formω,c (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 3/2 < σ < 2. 1 2 (R) was also proved under the same mass constraint. The problems Key words and phrases. derivative gDNLS, orbital instability, solitary wave solutions. Z. Guo is supported by ARC DP170101060.

On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions

“…Note that in (2.1) we only allow for a simple time-dependence exp(iωt) with ω = 1 in (2.1). This is not a restriction for the usual 2D NLS, given its scaling invariance, but it is a restriction for our model in which this invariance is broken (see also [13,27] for the connection between ω and the speed of stable solitary waves). For the classical NLS, i.e., ε = 0 and | δ | = 0, there exists a particular solution Q, called the nonlinear ground state, which is the unique radial and positive solution to (2.2), cf.…”

“…These numerical findings are consistent with analytical results for derivative NLS in one spatial dimension. For certain values of σ 1 and certain velosit v, the corresponding solitary wave solutions are found to be orbitally stable, see [8,13,27]. However, for general initial data and σ > 1 large enough, one expects finite-time blow-up, see [28].…”

“…Derivative NLS and their corresponding ground states are usually studied in one spatial dimension only, see e.g. [1,8,14,13,27,28,36,37] and references therein. For σ = 1, the classical one-dimensional derivative NLS is known to be completely integrable.…”

Microscopic conservation laws for the derivative Nonlinear Schrödinger equation

“…Miao, Tang and Xu use the structure analysis and classical variational argument to show the existence of solitary waves with two parameters and improve the global result of (1.1) in the energy space in [26], and further use perturbation argument, modulation analysis and Lyapunov stability to show the orbital stability of weak interaction multi-soliton solution with subcritical parameters in the energy space in [25]. We can also refer to [5,9,23,27,40] for the stability analysis of the solitary waves of the (generalized) derivative nonlinear Schrödinger equation in the energy space and to [10] for lower regularity result of (1.1) by almost conservation law in [41]. Since (1.1) is an integrable system in [1,18], there are lots of global wellposedness of (1.1) with mass restriction in the weighted Sobolev spapce based on the inverse scattering method, please refer to [15,16,17,32,33]and reference therein.…”

Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the endpoint case

“…Further, Fukaya [6] proved that the solitary waves solution u ω,c (t, x) is unstable when 7 6 < σ < 2, c = 2z 0 √ ω. Moreover, Guo, Ning and Wu [12] and Miao, Tang and Xu [22] independently proved that the solitary waves solution u ω,c (t, x) is unstable for any 1 < σ < 2 in borderline case c = 2z 0 √ ω. After these works, the stability theory when c = 2 √ ω, σ ∈ (1, 2) is unsolved.…”