Complex solitary waves and soliton trains in KdV and mKdV equations

We demonstrate the existence of complex solitary wave and periodic solutions of the Kortewegde Vries (KdV) and modified Korteweg-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 (P) and time-reversal (T) operations. The corresponding localized solitons are hydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishing intensity. The PT-odd complex soliton solution is shown to be iso-spectrally connected to the fundamental sech(2) solution through supersymmetry. Physically, these complex solutions are analogous to the experimentally observed grey solitons of non-liner Schrodinger equation, governing the dynamics of shallow water waves and hence may also find physical verification.