Solitons in a ${\cal PT}$- symmetric $χ^{(2)}$ coupler

Abstract: We consider the existence and stability of solitons in a $\chi^{(2)}$ coupler. Both the fundamental and second harmonics undergo gain in one of the coupler cores and are absorbed in the other one. The gain and losses are balanced creating a parity-time (${\cal PT}$) symmetric configuration. We present two types of families of ${\cal PT}$-symmetric solitons, having equal and different profiles of the fundamental and second harmonics. It is shown that gain and losses can stabilize solitons. Interaction of stable solitons is shown. In the cascading limit the model is reduced to the ${\cal PT}$-symmetric coupler with effective Kerr-type nonlinearity and balanced nonlinear gain and losses.