  The existence of soliton families in non-parity-time-symmetric complex
potentials remains poorly understood, especially in two spatial dimensions. In
this article, we analytically investigate the bifurcation of soliton families
from linear modes in one- and two-dimensional nonlinear Schr\"odinger equations
with localized Wadati-type non-parity-time-symmetric complex potentials. By
utilizing the conservation law of the underlying non-Hamiltonian wave system,
we convert the complex soliton equation into a new real system. For this new
real system, we perturbatively construct a continuous family of low-amplitude
solitons bifurcating from a linear eigenmode to all orders of the small soliton
amplitude. Hence, the emergence of soliton families in these
non-parity-time-symmetric complex potentials is analytically explained. We also
compare these analytically constructed soliton solutions with high-accuracy
numerical solutions in both one and two dimensions, and the asymptotic accuracy
of these perturbation solutions is confirmed.
