  We consider non-Abelian strings in N=2 supersymmetric QCD with the U$(N)$
gauge group and $N_f=N$ quark flavors deformed by a mass term for the adjoint
matter. This deformation breaks N=2 supersymmetry down to N=1. Dynamics of
orientational zero modes on the string world sheet are described then by
CP$(N-1)$ model with N=(0,2) supersymmetry. We study the string of a finite
length $L$ assuming compactification on a cylinder (periodic boundary
conditions). The world-sheet theory is solved in the large-$N$ approximation.
We find a rich phase structure in the $(L, \,u)$ plane where $u$ is a
deformation parameter. At large $L$ and intermediate $u$ we find a phase with
broken $Z_{2N}$ symmetry, $N$ vacua and a mass gap. At large values of $L$ and
$u$ still larger we have the $Z_{2N}$-symmetric phase with a single vacuum and
massless fermions. In both phases N=(0,2) supersymmetry is spontaneously
broken. We also observe a phase with broken SU$(N)\;$ symmetry at small $L$. In
the latter phase the mass gap vanishes and the vacuum energy is zero in the
leading $1/N$ approximation. However, we expect that $1/N$ corrections will
break N=(0,2) supersymmetry. We also discuss how this rich phase structure
matches the N=(2,2) limit in which the world-sheet theory has a single phase
with the mass gap independent of $L$.
