Instability of solitary wave solutions for derivative nonlinear Schrödinger equation in endpoint case (Q729921)

The authors of this very interesting paper, study the instability of solitary wave solutions for the derivative nonlinear Schrödinger (NLS) equation: \[ (\ast ) \;\;\;\;i\partial_tu+ \partial_{x}^2u+i\partial_x(|u|^2u)+b|u|^4u=0, \;\;t,x \in\mathbb{R}, \;\;b>0. \] This equation is a mathematical model of Alfvén waves in Plasma Physics and Nonlinear Optics. In the case \(b = 0\), by a suitable gauge transformation, the equation (\(\ast \)) can be transformed to the standard derivative nonlinear Schrödinger equation: \[ (\ast \ast ) \;\;\;\;i\partial_tu+ \partial_{x}^2u+i\partial_x(|u|^2u)=0. \] It is known that the Cauchy problem for (\(\ast \)) (also (\(\ast \ast \))) is locally well-posed in the energy space \(H^1 (\mathbb{R})\) (Sobolev space) provided that the initial data satisfies \(\|u_0\|_{L^2}<2\sqrt{\pi }\). Note that the solution of (\(\ast \)) satisfies three conservation laws: \(E(u(t))=E(u_0)\), \(P(u(t)) = P (u_0)\), \(M(u(t)) = M(u_0)\) for \(t\in [0,T_{\max })\), where \(E(u(t))=(1/2)\|\partial_xu\|_{L^2}^2-(1/4) (i|u|^2\partial_xu,u)_{L^2}-(b/6)\|u\|_{L^6}^6\), \(P(u(t))=(1/2)(i\partial_xu,u)_{L^2}\), \(M(u(t))=(1/2)\|u\|_{L^2}^2\). It is known that (\(\ast \ast \)) has a two-parameter family of solitary waves \(u_{\omega }(t, x)\) depending not only on \(t,x\) (time and space) but on a two dimensional vector \(\omega = (\omega_0,\omega_1)\in \mathbb{R}^{+}\times \mathbb{R}\), where \(\omega_1^2\leq 4\omega_0\). Next, the authors consider the stability theory for the equation (\(\ast \)) to the endpoint case \(\omega_{1} = 2\sqrt{\omega_{0}}\). Letting \(\omega_{1}=c>0\), \(\omega_0=c^2/4\) then the solitary wave solutions \(u_{\omega }(t, x)\) for this special case have the form \(R_c(t,x)=e^{ic^2t/4}\phi_c(x-ct)\) (\(c>0\)), and \(\phi_c(x)=e^{icx/2-(i/4) \int_{-\infty}^{x}|\varphi_c(\eta ) |^2d\eta }\varphi_c(x)\). Here \(\phi_c(x)\) and \(\varphi_c(x)\) are solutions of two different stationary differential equations. The main question concerns the instability of the solitary wave solutions in the above stated endpoint case with ``zero mass'' and a two-component \(b\). The main statement proved here is that ``if \(b \in (0, b_0)\) for some small \(b_0 > 0\), then the solitary wave solution \(R_c (t, x)\) of (\(\ast \)) is unstable''.