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Integral complex modified Korteweg-de Vries (Icm-KdV) equations

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  • Velasco-Juan, M.
  • Fujioka, J.

Abstract

In this article we examine different ways of introducing nonlocal effects in the complex modified Korteweg-de Vries (cmKdV) equation. Four possible nonlocal generalizations of the cmKdV equation are presented. In these variants of the cmKdV equation the nonlinear term has been replaced by one, two, or three integral terms. Three of these variants can be obtained by the least action principle, and they possess different equivalent Lagrangians. The relationship between these Lagrangians is entirely new. Therefore, a novel type of Lagrangian equivalence has been found. As Fubini´s theorem is needed to prove the equivalence of these Lagrangians, we call them “F-equivalent”. Noether´s theorem is applied to these integral equations, and different conserved quantities are obtained. The behavior of solitary waves in these models is also studied. It was found that one of these models has soliton-like solutions. Numerical solutions show that these solitons are stable, and the amplitudes of perturbed solitons attain constant values after a short transient period. Finally, a variational method was applied to get a better understanding of the behavior of these pulses during the transient periods.

Suggested Citation

  • Velasco-Juan, M. & Fujioka, J., 2020. "Integral complex modified Korteweg-de Vries (Icm-KdV) equations," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    • Handle: RePEc:eee:chsofr:v:134:y:2020:i:c:s0960077920300898
    DOI: 10.1016/j.chaos.2020.109687
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    References listed on IDEAS

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    1. Abdou, M.A., 2007. "The extended F-expansion method and its application for a class of nonlinear evolution equations," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 95-104.
    2. Zheng, Chun-Long & Fang, Jian-Ping & Chen, Li-Qun, 2005. "New variable separation excitations of (2+1)-dimensional dispersive long-water wave system obtained by an extended mapping approach," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1741-1748.
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    1. Flores-Calderón, R. & Fujioka, J. & Espinosa-Cerón, A., 2021. "Soliton dynamics of a high-density Bose-Einstein condensate subject to a time varying anharmonic trap," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).

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