Eur. Phys. J. B (2016) 89: 149
https://doi.org/10.1140/epjb/e2016-70130-7

Regular Article

Complex solitary waves and soliton trains in KdV and mKdV equations

¹ Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India
² BITS Pilani, K.K. Birla Goa Campus, Goa 403726, India

^a e-mail: panigrahi.iiser@gmail.com

Received: 1 March 2016
Received in final form: 20 April 2016
Published online: 13 June 2016

Abstract

We demonstrate the existence of complex solitary wave and periodic solutions of the Korteweg-de 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 sech² solution through supersymmetry. Physically, these complex solutions are analogous to the experimentally observed grey solitons of non-liner Schödinger equation, governing the dynamics of shallow water waves and hence may also find physical verification.