AI 中文总结
本文研究作为二分量第一修正KP层次的mToda层次，构造其τ函数与Lax算子的变换，推导ASvM公式及mToda加法公式，明确顶点算子作用与多步变换的等价性。
AI 中文摘要
本文研究修正Toda（mToda）层次，其可视为二分量第一修正Kadomtsev-Petviashvili（mKP）层次。首先探究Toda与mToda τ函数间的联系，在此基础上构造mToda τ函数与Lax算子的变换；进一步给出mToda平方本征函数对称性，推导Adler-Shiota-van Moerbeke（ASvM）公式，该公式在联系额外对称性对波函数的作用与τ函数的Sato-Bäcklund变换中起关键作用；最后通过建立顶点算子对mToda τ函数的作用与多步mToda变换的等价性，推导mToda加法公式，亦称广义Fay恒等式。
英文摘要
In this paper, we investigate the modified Toda (mToda) hierarchy, which can be regarded as the 2-component first modified Kadomtsev-Petviashvili (mKP) hierarchy. We first investigate the connection between the Toda and mToda tau functions. Based on this, we construct the transformations for the mToda tau functions and Lax operators. Furthermore, we present the mToda squared eigenfunction symmetries and derive the Adler-Shiota-van Moerbeke (ASvM) formula, which plays a crucial role by connecting the actions of the additional symmetries on the wave functions with the Sato--Bäcklund transformations of the tau functions. Finally, by establishing the equivalence between the actions of vertex operators on the mToda tau functions and the multi-step mToda transformations, we derive the mToda addition formulas, also known as the generalized Fay identities.
Comments28 pages
