  Recently, a number of nonlocal integrable equations, such as the PT-symmetric
nonlinear Schrodinger (NLS) equation and PT-symmetric Davey-Stewartson
equations, were proposed and studied. Here we show that many of such nonlocal
integrable equations can be converted to local integrable equations through
simple variable transformations. Examples include these nonlocal NLS and
Davey-Stewartson equations, a nonlocal derivative NLS equation, the reverse
space-time complex modified Korteweg-de Vries (CMKdV) equation, and many
others. These transformations not only establish immediately the integrability
of these nonlocal equations, but also allow us to construct their analytical
solutions from solutions of the local equations. These transformations can also
be used to derive new nonlocal integrable equations. As applications of these
transformations, we use them to derive rogue wave solutions for the partially
PT-symmetric Davey-Stewartson equations and the nonlocal derivative NLS
equation. In addition, we use them to derive multi-soliton and quasi-periodic
solutions in the reverse space-time CMKdV equation. Furthermore, we use them to
construct many new nonlocal integrable equations such as nonlocal short pulse
equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa-Satsuma
equations.
