{"paper":{"title":"Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schr\\\"odinger equation in the critical frequency case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cui Ning, Yifei Wu, Zihua Guo","submitted_at":"2018-03-21T00:19:38Z","abstract_excerpt":"We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\\\"odinger equation $$ i\\partial_{t}u+\\partial_{x}^{2}u+i|u|^{2\\sigma}\\partial_x u=0. $$ 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\\{ i\\frac c2 x-\\frac{i}{2\\sigma+2}\\int_{-\\infty}^{x}\\varphi^{2\\sigma}_{\\omega,c}(y)dy\\big\\}}. \\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 $-2\\sqrt{\\omega }