This paper is concerned with the construction of new solutions in terms of birational maps to the functional tetrahedron equation and parametric tetrahedron equation. We present a method for constructing solutions to the parametric tetrahedron equation via Darboux transformations. In particular, we study matrix refactorisation problems for Darboux transformations associated with the nonlinear Schrödinger (NLS) and the derivative nonlinear Schrödinger (DNLS) equation, and we construct novel nine-dimensional tetrahedron maps. We show that the latter can be restricted to six-dimensional parametric tetrahedron maps on invariant leaves. Finally, we construct parametric tetrahedron maps employing degenerated Darboux transformations of NLS and DNLS type.

本文关注的是根据功能四面体方程和参数四面体方程的双对映图构造新的解决方案。我们提出了一种通过Darboux变换构造参数四面体方程解的方法。特别是，我们研究了与非线性Schrödinger（NLS）和导数非线性Schrödinger（DNLS）方程有关的Darboux变换的矩阵重构问题，并构造了新颖的九维四面体图。我们表明后者可以限于不变叶上的六维参数四面体图。最后，我们使用NLS和DNLS类型的简并Darboux变换构造参数四面体图。