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

arXiv1804.02738MaRDI QIDQ6300078

Bing Li, Cui Ning

Publication date: 8 April 2018

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

Mathematics Subject Classification ID

Stability in context of PDEs (35B35) Second-order nonlinear hyperbolic equations (35L70)

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