Exact Rolling Tachyon in Noncommutative Field Theory

We study the exact rolling tachyon solutions in DBI type noncommutative field theory with a constant open string metric and noncommutative parameter on an unstable D$p$-brane. Functional shapes of the obtained solutions span all possible homogeneous rolling tachyon configurations; that is, they are hyperbolic-cosine, hyperbolic-sine, and exponential under $1\mathrm{/}\cosh$ runaway NC tachyon potential. Even if general DBI type NC electric field is turned on, only a constant electric field satisfies the equations of motion, and again, exact rolling tachyon solutions are obtained.