Title: Controlling of blow-up responses by a nonlinear $\cal{PT}$ symmetric coupling,
Abstract: We investigate the dynamics of a coupled waveguide system with competing
linear and nonlinear loss-gain profiles which can facilitate power saturation.
We show the usefulness of the model in achieving unidirectional beam
propagation. In this regard, the considered type of coupled waveguide system
has two drawbacks, (i) difficulty in achieving perfect isolation of light in a
waveguide and (ii) existence of blow-up type behavior for certain input power
situations. We here show a nonlinear $\cal{PT}$ symmetric coupling that helps
to overcome these two drawbacks. Such a nonlinear coupling has close connection
with the phenomenon of stimulated Raman scattering. In particular, we have
elucidated the role of this nonlinear coupling using an integrable $\cal{PT}$
symmetric situation. In particular, using the integrals of motion, we have
reduced this coupled waveguide problem to the problem of dynamics of a particle
in a potential. With the latter picture, we have clearly illustrated the role
of the considered nonlinear coupling. The above $\cal{PT}$ symmetric case
corresponds to a limiting form of a general equation describing the phenomenon
of stimulated Raman scattering. We also point out the ability to transport
light unidirectionally even in this general case.