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Continuous families of solitons in generalized nonlinear Sch\"odingerequations with non-PT-symmetric complex potentials are studied analytically.Under a weak assumption, it is shown that stationary equations for solitonsadmit a constant of motion if and only if the complex potential is of a specialform $g^2(x)+ig'(x)$, where $g(x)$ is an arbitrary real function. Using thisconstant of motion, the second-order complex soliton equation is reduced to anew second-order real equation for the amplitude of the soliton. From this realsoliton equation, a novel perturbation technique is employed to show thatcontinuous families of solitons always bifurcate out from linear discrete modesin these non-PT-symmetric complex potentials. All analytical results arecorroborated by numerical examples.

Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials