The nonstandard constrained KP hierarchy and the generalized Miura transformations

Miura and Generalized Bäcklund Transformation for KdV Hierarchy

Using the fact that Miura transformation can be expressed in the form of gauge transformation connecting the KdV and mKdV equations, we discuss the derivation of the Bäcklund transformation and its Miura-gauge transformation connecting both hierarchies.

Regular solution and lattice miura transformation of bigraded Toda Hierarchy

In this paper, we give finite dimensional exponential solutions of the bigraded Toda Hierarchy(BTH). As an specific example of exponential solutions of the BTH, we consider a regular solution for the $(1,2)$-BTH with $3\times 3$-sized Lax matrix, and discuss some geometric structure of the solution from which the difference between $(1,2)$-BTH and original Toda hierarchy is shown. After this, we construct another kind of Lax representation of $(N,1)$-bigraded Toda hierarchy($(N,1)$-BTH) which does not use the fractional operator of Lax operator. Then we introduce lattice Miura transformation o…

Gauge Miura and Backlund Transformations for Generalized $A_n$-KdV Hierarchies

The construction of Miura and Bäcklund transformations for $A_n$ mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in the well known $sl(2)$ case, we derive and relate the equations of motion for the two hierarchies. Moreover, the Miura-gauge transformation is not unique, instead, it is shown to be connected to a set of generators labeled by the exponents of $A_n$ The construction of generalized gauge-Bäcklund transformation for the $A_n$-KdV hierarchy is obtained as a composition of Miura and Bäcklund-gauge transformatio…

双微分演算中的Miura变换及Fokas-Lenells方程的向量Darboux变换

Miura transformation in bidifferential calculus and a vectorial Darboux transformation for the Fokas-Lenells equation

本文利用双微分演算导出耦合Fokas-Lenells系统的向量Darboux变换，并构造Miura变换及平面波种子精确解。

修正Toda层级

The Modified Toda Hierarchy

本研究在Sato理论框架下从二分量第一修正KP层级的双线性方程出发构建了修正Toda（mToda）层级，证明其与Lax公式等价、与Toda层级存在Miura联系，并借此构造了两类层级的Darboux变换。

规范-米乌拉变换下的复KdV怪波

Complex KdV rogue waves from gauge-Miura transformation

本文利用规范-米乌拉对应关系，将KdV层级解分类为mKdV层级的基本tau函数，并解析描述了复KdV系统中的怪波现象。