  We demonstrate the existence of complex solitary wave and periodic solutions
of the Kortweg de-vries (KdV) and modified Kortweg de-Vries (mKdV) equations.
The solutions of the KdV (mKdV) equation appear in complex-conjugate pairs and
are even (odd) under the simultaneous actions of parity ($\cal{P}$) and
time-reversal ($\cal{T}$) operations. The corresponding localized solitons are
hydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically
vanishing intensity. The $\cal{PT}$-odd complex soliton solution is shown to be
iso-spectrally connected to the fundamental $sech^2$ solution through
supersymmetry.
