  The Korteweg-de Vries equation u_t+uu_x+u_{xxx}=0 is PT symmetric (invariant
under space-time reflection). Therefore, it can be generalized and extended
into the complex domain in such a way as to preserve the PT symmetry. The
result is the family of complex nonlinear wave equations u_t-iu(i
u_x)^epsilon+u_{xxx}=0, where epsilon is real. The features of these equations
are discussed. Special attention is given to the epsilon=3 equation, for which
conservation laws are derived and solitary waves are investigated.
