  In this paper, by considering the potential application in two mode nonlinear
waves in nonlinear fibers under a certain case, we define a coupled nonlinear
Schr\"odinger equation(called Frobenius NLS equation) including its Lax pair.
Afterwards, we construct the Darboux transformations of the Frobenius NLS
equation. New solutions can be obtained from known seed solutions by the
Dardoux transformations. Soliton solutions are generated from trivial seed
solutions. Also we derive breather solutions $q,r$ of the Frobenius NLS
equation obtained from periodic seed solutions. Interesting enough, we find the
amplitudes of $r$ vary in size in different areas with period-like fluctuations
in the background. This is very different from the solution $q$ of the
single-component classical nonlinear Schr\"odinger equation. Then, the rogue
waves of the Frobenius NLS equation are given explicitly by a Taylor series
expansion about the breather solutions $q,r$. Also the graph of rogue wave
solution $r$ shows that the rogue wave has fluctuations around the peak. The
reason of this phenomenon should be in the dynamical dependence of $r$ on $q$
which is independent on $r$.
