Complex Solitary Waves and Soliton Trains in KdV and mKdV Equations

摘要

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