BPS equations for a one-scalar D3-brane ($\mathbf{E}=\cos \theta \nabla X,\mathbf{B}=\sin \theta \nabla X,\Delta X=0$)

For one constant charge angle, the DBI Hamiltonian bound for an electric spike and its dyonic extension give the displayed equations away from sources. Each aligned branch preserves eight of the sixteen linearly realized worldvolume supersymmetries. The corresponding spacetime string–D3 configuration preserves eight of thirty-two. The bound and charge interpretation are derived in BPS Bounds for Worldvolume Branes.