  With the aim at understanding the non-monotonic $d_{x^{2}-y^{2}}$-wave gap,
we analyze the local electronic structure near impurities in the electron-doped
cuprate superconductors. We find that the local density of states near a
non-magnetic impurity in the scenario of $d_{x^{2}-y^{2}}$-wave
superconductivity with higher harmonics is qualitatively different from that
obtained from the $d_{x^{2}-y^{2}}$-wave superconductivity coexisting with
antiferromagnetic spin density wave order. We propose that spin-polarized
scanning tunneling microscopy measurements can distinguish the two scenarios
and shed light on the real physical origin of a non-monotonic
$d_{x^{2}-y^{2}}$-wave gap.
