  The Khuri-Treiman formalism models the partial-wave expansion of a scattering
amplitude as a sum of three individual truncated series, capturing the
low-energy dynamics of the direct and cross channels. We cast this formalism
into dispersive equations to study $\pi\pi$ scattering, and compare their
expressions and numerical output to the Roy and GKPY equations. We prove that
the Khuri-Treiman equations and Roy equations coincide when both are truncated
to include only $S$- and $P$-waves. When higher partial waves are included, we
find an excellent agreement between the Khuri-Treiman and the GKPY results.
This lends credence to the notion that the Khuri-Treiman formalism is a
reliable low-energy tool for studying hadronic reaction amplitudes.
