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We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrodinger equation $$ ipartial_{t}u+partial_{x}^{2}u+i|u|^{2sigma}partial_x u=0. $$ The equation has a two-parameter family of solitary wave solutions of the form begin{align*} phi_{omega,c}(x)=varphi_{omega,c}(x)exp{big{ ifrac c2 x-frac{i}{2sigma+2}int_{-infty}^{x}varphi^{2sigma}_{omega,c}(y)dybig}}. end{align*} Here $ varphi_{omega,c}$ is some real-valued function. It was proved in cite{LiSiSu1} that the solitary wave solutions are stable if $-2sqrt{omega }<c <2z_0sqrt{omega }$, and unstable if $2z_0sqrt{omega }<c <2sqrt{omega }$ for some $z_0in(0,1)$. We prove the instability at the borderline case $c =2z_0sqrt{omega }$ for $1<sigma<2$, improving the previous results in cite{Fu-16-DNLS} where $3/2<sigma<2$.
In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesnt enjoy the Galilean invariance any more, the structure analysis here is closely related
An explicit lifespan estimate is presented for the derivative Schrodinger equations with periodic boundary condition.
We consider the derivative nonlinear Schrodinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we discuss is
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