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Classification of integrable KdV equation with nonlocal terms

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  • Zhao, Min

Abstract

Nonlocal integrable equations have attracted considerable research interest in recent decades, providing critical insights into diverse physical systems ranging from stratified fluids to quantum field theories. In this paper, we study the classification of nonlocal KdV equations using the Perturbative symmetry approach. One of the resulting equations is shown to stem from the compatibility conditions of a Lax pair system of linear PDEs, thereby establishing its integrability. We explicitly construct its N-pulse solutions and rigorously demonstrate that the pole dynamics of these solutions correspond precisely to the integrable Calogero-Moser system. Furthermore, unlike the classical real-valued KdV equation, the nonlocal KdV equation shares key properties with the complex-valued KdV equation with abundant finite time blow-up solutions.

Suggested Citation

  • Zhao, Min, 2026. "Classification of integrable KdV equation with nonlocal terms," Chaos, Solitons & Fractals, Elsevier, vol. 202(P2).
    • Handle: RePEc:eee:chsofr:v:202:y:2026:i:p2:s0960077925016431
    DOI: 10.1016/j.chaos.2025.117630
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    References listed on IDEAS

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    1. Velasco-Juan, M. & Fujioka, J., 2020. "Integral complex modified Korteweg-de Vries (Icm-KdV) equations," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    2. Flamarion, Marcelo V. & Pelinovsky, Efim, 2024. "Solitons in dissipative systems subjected to random force within the Benjamin–Ono type equation," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    3. Pawan Negi & Trilochan Sahoo & Niharika Singh & Yury Stepanyants, 2023. "Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow," Mathematics, MDPI, vol. 11(15), pages 1-15, August.
    4. Han, Peng-Fei & Zhang, Yi, 2024. "Investigation of shallow water waves near the coast or in lake environments via the KdV–Calogero–Bogoyavlenskii–Schiff equation," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    5. Flamarion, Marcelo V. & Pelinovsky, Efim, 2024. "Auto-resonance process under the interaction of solitons with external force and dissipation," Chaos, Solitons & Fractals, Elsevier, vol. 189(P2).
    6. Li, Y. Charles, 2009. "Simple explicit formulae for finite time blow up solutions to the complex KdV equation," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 369-372.
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