We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and nonlocal Korteweg-de Vries equation and elaborate on how multi-soliton solutions with various types of novel qualitative behaviour can be constructed. Corresponding to the different multicomplex units in these extensions, real, hyperbolic or imaginary, the wave equations and their solutions exhibit multiple versions of antilinear or 𝒫𝒯-symmetries. Utilizing these symmetries forces certain components of the conserved quantities to vanish, so that one may enforce them to be real. We find that symmetrizing the noncommutative equations is equivalent to imposing a 𝒫𝒯-symmetry for a newly defined imaginary unit from combinations of imaginary and hyperbolic units in the canonical representation.

中文翻译：

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多重孤子

我们讨论了具有多孤子解的实非线性波动方程的可积扩展，到它们的双复数、四元数、共四元数和八元数形式。特别是，我们研究了局部和非局部 Korteweg-de Vries 方程的这些变体，并详细说明了如何构建具有各种新型定性行为的多孤子解。对应于这些扩展中的不同多重复形单元，实数、双曲线或虚数，波动方程及其解表现出多种形式的反线性或 𝒫𝒯 对称性。利用这些对称性迫使守恒量的某些分量消失，以便人们可以强制它们为实数。