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There are two cases when the nonlinear Schrödinger equation with an external complex potential is well known to support continuous families of localized stationary modes: the   PT  ‐symmetric potentials and the Wadati potentials. Recently, Kominis et al. have suggested that the continuous families can be also found in complex potentials of the form    W   (  x  )   =   W  1    (  x  )   +  i  C   W   1  ,  x(  x  )    , where   C   is an arbitrary real andW  1    (  x  )is a real‐valued and bounded differentiable function. Here we study in detail nonlinear stationary modes that emerge in complex potentials of this type (for brevity, we call them  W‐dW potentials ). First, we assume that the potential is small and employ asymptotic methods to construct a family of nonlinear modes. Our asymptotic procedure stops at the terms of the    ε  2    order, where small   ε   characterizes amplitude of the potential. We therefore conjecture that no continuous families of authentic nonlinear modes exist in this case, but “pseudo‐modes” that satisfy the equation up to    ε  2   ‐error can indeed be found in W‐dW potentials. Second, we consider the particular case of a W‐dW potential well of finite depth and support our hypothesis with qualitative and numerical arguments. Third, we simulate the nonlinear dynamics of found pseudo‐modes and observe that, if the amplitude of W‐dW potential is small, then the pseudo‐modes are robust and display persistent oscillations around a certain position predicted by the asymptotic expansion. Finally, we study the authentic stationary modes that do not form a continuous family, but exist as isolated points. Numerical simulations reveal dynamical instability of these solutions.

On nonexistence of continuous families of stationary nonlinear modes for a class of complex potentials