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Complex solitary waves and soliton trains in KdV and mKdV equations

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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.

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Authors and Affiliations

  1. Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India

    Subhrajit Modak & Prasanta Kumar Panigrahi

  2. BITS Pilani, K.K. Birla Goa Campus, Goa, 403726, India

    Akhil Pratap Singh

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  1. Subhrajit Modak
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  2. Akhil Pratap Singh
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  3. Prasanta Kumar Panigrahi
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Correspondence to Prasanta Kumar Panigrahi.

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Modak, S., Singh, A. & Panigrahi, P. Complex solitary waves and soliton trains in KdV and mKdV equations. Eur. Phys. J. B 89, 149 (2016). https://doi.org/10.1140/epjb/e2016-70130-7

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