General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the $PT$-symmetric system

With the stationary solution assumption, we establish the connection between the nonlocal nonlinear Schrödinger (NNLS) equation and an elliptic equation. Then, we obtain the general stationary solutions and discuss the relevance of their smoothness and boundedness to some integral constants. Those solutions, which cover the known results in the literature, include the unbounded Jacobi elliptic-function and hyperbolic-function solutions, the bounded sn-, cn-, and dn-function solutions, as well as the hyperbolic soliton solutions. By the imaginary translation transformation of the NNLS equation, we also derive the complex-amplitude stationary solutions, in which all the bounded cases obey either the $PT$- or anti-$PT$-symmetric relation. In particular, the complex tanh-function solution can exhibit no spatial localization in addition to the dark- and antidark-soliton profiles, which is in sharp contrast with the common dark soliton. Considering the physical relevance to the $PT$-symmetric system, we show that the complex-amplitude stationary solutions can yield a wide class of complex and time-independent $PT$-symmetric potentials, and the symmetry breaking does not occur in the $PT$-symmetric linear system with the associated potentials.