Exact representations for coupled bright and dark solitary waves of quadratically nonlinear systems

Abstract

A thorough search is performed for all cases in which solutions of the quadratically coupled nonlinear equations for χ(2) cascading may be expressed in terms of hyperbolic functions. This reveals two new classes (IV and VI) in addition to classes I–III, which have appeared in many recent papers. These new solutions describe pulses with a nonzero cw background in one or both modes. They include bright–dark, brighter–brighter, twin-hole, and dark–dark solutions, and each contains an adjustable parameter related to the phase shift across the pulse. For a planar waveguide, any of the class VI profiles may arise for any choice of pulse orientation parameters κJ giving pp̃>0, whereas any of the class IV profiles may arise if pp̃<0. In each case, the choice then determines the remaining pulse parameters in the exact representation.