多复孤子
Multicomplex solitons
摘要：
本文讨论了具有多孤子解的实非线性波方程的可积推广，得到了它们的双复、四元数、余四元数和八元数形式。特别是，我们调查这些变量的本地和非本地Korteweg-de Vries方程和详细说明如何多孤子解决方案与各种类型的新的定性行为可以构建。对应于这些扩展中的不同复单位，实的，双曲的或虚的，波动方程及其解表现出多个版本的反线性或$\mathcal{PT}$ -对称性。利用这些对称性可以迫使守恒量的某些分量为零，这样就可以强制它们为实数。我们发现，对称化的非交换方程是等价于强加一个$\mathcal{PT}$ -对称性的一个新定义的虚单位从虚和双曲单位的组合在规范表示。
英文摘要：
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 $\mathcal{PT}$-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 $\mathcal{PT}$-symmetry for a newly defined imaginary unit from combinations of imaginary and hyperbolic units in the canonical representation.
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