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arXiv · 1804.02738

Instability of the solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation in the endpoint case

Bing Li·Cui Ning
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Abstract

We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0, $$ where $1<\sigma<2$. The equation has a two-parameter family of solitary wave solutions of the form $$ u_{\omega,c}(t,x)=e^{i\omega t+i\frac c2(x-ct)-\frac{i}{2\sigma+2}\int_{-\infty}^{x-ct}\varphi^{2\sigma}_{\omega,c}(y)dy}\varphi_{\omega,c}(x-ct). $$ The stability theory in the frequency region of $|c|<2\sqrt{\omega}$ was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case $c=2\sqrt{\omega}$.

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Bing Li, Cui Ning. 2018-04-08. Instability of the solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation in the endpoint case. https://arxiv.org/abs/1804.02738

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