Exact solitonic solutions for optical media with χ(2) nonlinearity and PT-symmetric potentials

We study the χ(2) system of equations with the PT-symmetric potentials. We obtain exact solitary wave solutions for this system for some forms of the PT-symmetric potentials.

with d 1,2 are diffraction coefficients, q is the phase missmatch. Here V 1,2 (x) are real parts of the refraction index. PT-symmetry impose restriction on V 1,2 to be even functions of x and and on the imaginary potential W 1,2 (x) describing the inhomogeneous in space gain/loss, to be odd functions of x. Introducing representation u(x, z) = U (x)e −iωz , v(x, z) = V (x)e −2iωz , we obtain the system of equations for U (x) and W (x) where σ = 2ω + q. It is useful to introduce the amplitude-phase variables by Then we obtain the system: To find the exact solutions of the system (5) we will apply the inverse engineering approach i.e. construct the potentials admitting the exact solutions [17,7].

Bright soliton solutions
First we consider the case of localized solutions in the form of bright-bright solitonic solutions.
1. Let look for solutions of the form: This form, for constant parameters case, has been introduced by Karamzin-Sukhorukov [18]. For the phase we impose the form: Then we obtain from (5) for the W 1 (x) Analogously we have found We obtain the relation between W 01 , W 02 From equations for amplitudes we have ).
The phase is: For example if r = 1, d 1 = 1/2, d 2 = 1/4 we obtain that the phase mismatch should be q = 3. Taking into account the condition A 2 > 0, dynamical breaking of the PT-symmetry occurs, when B > 0 and the strength of imaginary part of potential excess the value or when B < 0 and with V 01 = 2V 02 , d 1 = 2d 2 . Existence and stability of the solution when the conditions above are not fulfilled requires a separate consideration. 2. Second choice of the solution is: This form for constant parameters of χ (2) system has been introduced in [19]. Again we will assume the functional dependence for the phase gradients in the form (10). Then we obtain for W i (x) relations: W 1 = 3Cd 1 rsech(rx) tanh(rx) = W 01 sech(rx) tanh(rx), W 2 = 10Cd 2 rsech(rx) tanh(rx) = W 02 sech(rx) tanh(rx), It can be shown, that ω = −d 1 r 2 .
For soliton amplitudes we find respectively: For real potentials we have expressions where Again we have the dynamical breaking of the PT symmetry above threshold value for the strength of the imaginary part of potential W It is useful to consider the large negative mismatch case when |q| >> 1. Then v ≈ −u 2 /|q| and the system is reduced to the PT-symmetric extension of the single NLS equation(the socalled a cascading limit case) The solitonic solution for the potentials V = V 0 sech 2 (x), W = W 0 sech(x)tanh(x) is well known [6] and coincides with the solution (19), when |q| >> 1, r = 1, d 1 = 1. 3.Third choice of solutions is: This type of solitonic solutions for an homogeneous χ (2) media is considered in [20,21]. Let take the phase of the form θ x = Csech(rx) tanh(rx). Looking for the imaginary part of potential of the form Also we find 1 ) + 6d 1 r 2 ), 2 ).
So, as one can note, this is the example of exact solution in the complex non PT symmetric potential, since both real and imaginary parts of the potential are even functions of the x variable.

Conclusion
. We have found exact solitary solutions for quadratically nonlinear media with PT-symmetric potentials. The solutions were in the form of bright-bright solitons. Also the example of exact solution in non PT symmetric potential is given. The investigation of stability of obtained exact solutions is important problem and requires a separate analysis.