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The symmetry breaking of solitons in the nonlinear Schrödinger equation with cubic–quintic competing nonlinearity and parity-time symmetric potential is studied. At first, a new asymmetric branch separates from the fundamental symmetric soliton at the first power critical point, and then, the asymmetric branch passes through the branch of the fundamental symmetric soliton and finally merges into the branch of the fundamental symmetric soliton at the second power critical point, while the power of the soliton increases. This leads to the symmetry breaking and double-loop bifurcation of fundamental symmetric solitons. From the power-propagation constant curves of solitons, symmetric fundamental and tripole solitons, asymmetric solitons can also exist. The stability of symmetric fundamental solitons, asymmetric solitons, and symmetric tripole solitons is discussed by the linear stability analysis and direct simulation. Results indicate that symmetric fundamental solitons and symmetric tripole solitons tend to be stable with the increase in the soliton power. Asymmetric solitons are unstable in both high and low power regions. Moreover, with the increase in saturable nonlinearity, the stability region of fundamental symmetric solitons and symmetric tripole solitons becomes wider.

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The symmetry breaking phenomenon of the parity-time (PT) symmetric solitons in self-defocusing saturable nonlinear Schrödinger equation is studied. As the soliton power increases, branches of asymmetric solitons are separated from antisymmetric solitons, and they coexist with both symmetric and antisymmetric solitons. The anti-symmetric solitons require different power thresholds when they are under different saturable nonlinear strength. The stronger the saturable nonlinearity is, the larger the power threshold is. The saturable nonlinear strength has obvious modulation effect on the symmetry breaking of antisymmetric solitons and the bifurcation of the power curve. However, when the modulation strength of PT- symmetric potential increases, the effect of this modulation effect weakens. The antisymmetric solitons are only stable in the low power region, and the stability of symmetric and asymmetric solitons is less affected by the soliton power. The increase of the saturable nonlinear strength leads to the increase of the critical power of the symmetry breaking. When a beam propagates in a PT-symmetric optical waveguide, the symmetry breaking of antisymmetric solitons can be controlled by changing the saturable nonlinear strength.