  The instability of a streak and its nonlinear evolution are investigated by
direct numerical simulation (DNS) for plane Poiseuille flow at Re=3000. It is
suggested that there exists a traveling-wave solution (TWS). The TWS is
localized around one of the two walls and notably resemble to the coherent
structures observed in experiments and DNS so far. The phase space structure
around this TWS is similar to a saddle point. Since the stable manifold of this
TWS is extended close to the quasi two dimensional (Q2D) energy axis, the
approaching process toward the TWS along the stable manifold is approximately
described as the instability of the streak (Q2D flow) and the succeeding
nonlinear evolution. Bursting corresponds to the escape from the TWS along the
unstable manifold. These manifolds constitute part of basin boundary of the
turbulent state.
