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

Analysis of PDEs