Reflection solitons supported by competing nonlinear gratings

We analytically investigate solitons counterpropagating in a longitudinally modulated Kerr medium by means of the Hirota bilinear approach. We show that Hirota solvability (partial integrability of the system) physically corresponds to the exact mutual cancellation of the effects of all the underlying nonlinear gratings. This cancellation is achieved in two different situation: either through a suitable choice of the field profile or by means of a peculiar tailoring of the nonlinear modulation. In the first situation we obtain both bright and dark one-soliton solutions whose intensity ratio between forward and backward propagating beams is set by the nonlinear modulation. In the second situation, we derive two-soliton solutions obtained by nonlinearly dressing two independent linear grating eigenmodes sharing the same propagation constant.