In this paper, we firstly construct a weakly coupled Toda lattices with indefinite metrics which consist of 2 N different coupled Hamiltonian systems. Afterwards, we consider the iso-spectral manifolds of extended tridiagonal Hessenberg matrix with indefinite metrics which is an extension of a strict tridiagonal matrix with indefinite metrics. For the initial value problem of the extended symmetric Toda hierarchy with indefinite metrics, we introduce the inverse scattering procedure in terms of eigenvalues by using the Kodama’s method. In this article, according to the orthogonalization procedure of Szegö, the relationship between the $$\tau $$ τ -function and the given Lax matrix is also discussed. We can verify the results derived from the orthogonalization procedure with a simple example. After that, we construct a new strongly coupled Toda lattices with indefinite metrics and derived its tau structures. At last, we generalize the weakly coupled Toda lattices with indefinite metrics to the $$Z_{n}$$ Z n -Toda lattices with indefinite metrics.

中文翻译：

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具有不定度量的有限非周期 Toda 格的扩展

在本文中，我们首先构造了一个由 2 N 个不同的耦合哈密顿系统组成的具有不确定度量的弱耦合 Toda 格。然后，我们考虑具有不确定度量的扩展三对角 Hessenberg 矩阵的等谱流形，它是具有不确定度量的严格三对角矩阵的扩展。对于具有不确定度量的扩展对称 Toda 层次的初值问题，我们通过使用 Kodama 方法引入了特征值方面的逆散射过程。在本文中，根据 Szegö 的正交化程序，还讨论了 $$\tau $$ τ 函数与给定的 Lax 矩阵之间的关系。我们可以通过一个简单的例子来验证从正交化过程中得出的结果。之后，我们构建了一个新的具有不确定度量的强耦合 Toda 格并导出了它的 tau 结构。最后，我们将具有不确定度量的弱耦合 Toda 格推广到具有不确定度量的 $$Z_{n}$$Z n -Toda 格。