  We consider the electromagnetic waves propagating in the system of coupled
waveguides. One of the system components is a standard waveguide fabricated
from nonlinear medium having positive refraction and another component is a
waveguide produced from an artificial material having negative refraction. The
metamaterial constituting the second waveguide has linear characteristics and a
wave propagating in the waveguide of this type propagates in the direction
opposite to direction of energy flux. 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.

Pattern Formation and Solitons