Potential well theory for the derivative nonlinear Schrödinger equation

We consider the following nonlinear Schr\"{o}dinger equation of derivative type: \begin{equation}i \partial_t u + \partial_x^2 +i |u|^{2} \partial_x +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this is known as a gauge equivalent form well-known (DNLS), which mass critical and completely integrable. The can be considered generalized DNLS while preserving criticality Hamiltonian structure. For it that if initial data $u_0\in H^1(\mathbb{R})$ satisfies condition $\| u_0\|_{L^2}^2 <4\pi$, corresponding solution global bounded. In paper we first establish on for general $b\in\mathbb{R}$, exactly to $4\pi$-mass DNLS, then characterize from viewpoint potential well theory. see threshold value gives turning point in structure wells generated by solitons. particular, our results give characterization both algebraic