  We have studied the weakly non-linear quantum transport properties of a
two-dimensional quantum wire which can be solved exactly. The non-linear
transport coefficients have been calculated and interesting physical properties
revealed. In particular we found that as the incoming electron energy
approaches a resonant point given by energy $E=E_r$, where the transport is
characterized by a complete reflection, the second order non-linear conductance
changes its sign. This has interesting implications to the current-voltage
characteristics. We have also investigated the establishment of the gauge
invariance condition. We found that for systems with a finite scattering
region, correction terms to the theoretical formalism are needed to preserve
the gauge invariance. These corrections were derived analytically for this
model.
