  We consider the following nonlinear Schr\"{o}dinger equation with derivative:
\begin{equation} iu_t =-u_{xx} -i |u|^{2}u_x -b|u|^4u , \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 the well-known derivative nonlinear
Schr\"{o}dinger (DNLS) equation. The soliton profile of DNLS satisfies a
certain double power elliptic equation with cubic-quintic nonlinearities. The
quintic nonlinearity in our equation only affects the coefficient in front of
the quintic term in the elliptic equation, so in this sense the additional
nonlinearity is natural as a perturbation preserving soliton profiles of DNLS.
When $b\ge 0$, the equation has degenerate solitons whose momentum and energy
are zero, and if $b=0$, they are algebraic solitons. Inspired from the works on
instability theory of the $L^2$-critical generalized KdV equation, we study the
instability of degenerate solitons in a qualitative way, and when $b>0$, we
obtain a large set of initial data yielding the instability. The arguments
except one step in our proof work for the case $b=0$ in exactly the same way,
which is a small step towards understanding the dynamics around algebraic
solitons of the DNLS equation.
