Solitary electromagnetic wave propagation in the asymmetric oppositely directed coupler

We consider electromagnetic waves propagating in the system of coupled waveguides. One of the system components is a standard waveguide fabricated from nonlinear material having positive refraction index and another component is a waveguide produced from an artificial material having negative refraction index. In this waveguide a wave propagates in the direction opposite to direction of energy flux. We suppose that constituting metamaterial has linear properties. It is found that the coupled nonlinear solitary waves propagating both in the same direction are exist in this oppositely directed coupler due to linear coupling between nonlinear positive refractive waveguide and linear negative refractive waveguide. The corresponding analytical solution is found and it is used for numerical simulation to illustrate that the results of the solitary wave collisions are sensible to the relative velocity of the colliding solitary waves. The effect of linear losses in negative refractive waveguide is considered numerically.