Instantons in the U(1) Born-Infeld Theory and Noncommutative Gauge Theory

We derive a BPS-type bound for four-dimensional Born-Infeld action with constant B field background. The supersymmetric configuration saturates this bound and is regarded as an analog of instanton in U(1) gauge theory. Furthermore, we find the explicit solutions of this BPS condition. These solutions have a finite action proportional to the instanton number and represent D(p-4)-branes within a Dp-brane although they have a singularity at the origin. Some relations to the noncommutative U(1) instanton are discussed.