  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 a gauge equivalent form of well-known
derivative nonlinear Schr\"{o}dinger (DNLS) equation. The soliton profile of
the DNLS equation satisfies a certain double power elliptic equation with
cubic-quintic nonlinearities. The quintic nonlinearity in the equation only
affects the coefficient in front of the quintic term in the elliptic equation,
so the additional nonlinearity is natural as a perturbation preserving soliton
profiles of the DNLS equation. If $b>-\frac{3}{16}$, the equation has algebraic
solitons as well as exponentially decaying solitons. In this paper we study
stability properties of solitons by variational approach, and prove that if
$b<0$, all solitons including algebraic solitons are stable in the energy
space. The existence of stable algebraic solitons shows an interesting
mathematical example because stable algebraic solitons are not known in the
context of the corresponding double power NLS.

Analysis of PDEs