We propose a scheme comprising an array of anisotropic optical waveguides, embedded in a gas of cold atoms, which can be tuned from a Hermitian to an odd-$\mathcal{P}\mathcal{T}$-symmetric configuration through the manipulation of control and assistant laser fields. We show that the system can be controlled by tuning intra- and intercell coupling coefficients, enabling the creation of topologically distinct phases and linear topological edge states. The waveguide array, characterized by a quadrimer primitive cell, allows for implementing transitions between Hermitian and odd-$\mathcal{P}\mathcal{T}$-symmetric configurations, broken and unbroken $\mathcal{P}\mathcal{T}$-symmetric phases, topologically trivial and nontrivial phases, as well as transitions between linear and nonlinear regimes. The introduced scheme generalizes the Rice-Mele Hamiltonian for a nonlinear non-Hermitian quadrimer array featuring odd-$\mathcal{P}\mathcal{T}$ symmetry and makes accessible unique phenomena and functionalities that emerge from the interplay of non-Hermiticity, topology, and nonlinearity. We also show that in the presence of nonlinearity the system sustains nonlinear topological edge states bifurcating from the linear topological edge states and the modes without a linear limit. Each nonlinear mode represents a doublet of odd-$\mathcal{P}\mathcal{T}$-conjugate states. In the broken $\mathcal{P}\mathcal{T}$ phase, the nonlinear edge states may be effectively stabilized when an additional absorption is introduced into the system.

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• Edge states
• Nonlinear beam dynamics
• Topological phase transition
• PT-symmetry