Complex Solitary Waves and Soliton Trains in KdV and mKdV Equations

Abstract

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 ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) operations. The corresponding localized solitons are hydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishing intensity. The $\mathcal{P}\mathcal{T}$-odd complex soliton solution is shown to be iso-spectrally connected to the fundamental $sec{h}^2$ solution through supersymmetry.

References in corpus (2)

• Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
• Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations

Cited by in corpus (1)

• Exact $P_s{T}_d$ Invariant and $P_s{T}_d$ Symmetric Breaking solutions, Symmetry Reductions and Bäcklund Transformations For An AB-KdV System