Kh. S. Nirov, A. V. Razumov
We consider abelian twisted loop Toda equations associated with the complex general linear groups. The Dodd--Bullough--Mikhailov equation is a simplest particular case of the equations under consideration. We construct new soliton solutions of these Toda equations using the method of rational dressing.
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We present a systematic and detailed review of the application of the method of Hirota and the rational dressing method to abelian Toda systems associated with the untwisted loop groups of complex general linear groups. Emphasizing the rational dressing method, we compare the soliton solutions constructed within these two approaches, and show that the solutions obtained by the Hirota's method are a subset of those obtained by the rational dressing method.
Matrix Dressing Beyond Pfaff-Toda
本文提出矩阵伪差分算子框架统一Adler-Pfaff与Pfaff-Toda层级，引入矩阵Pfaff-Toda层级，恢复连续与多分量情形，并导出相关tau函数与约化结果。
Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. A technical clue of their method is a pair of dressing operators on the Fock space of 2D charged free fermion fields. We reformulate these operators as difference operators in the Lax formalism of the 2D Toda hierarchy. This leads to a new explanation to the question of why the equivariant Toda hierarchy emerges in the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. Moreover, the non-equivariant limit of these operators turns out to capture the integrable …
We construct soliton solutions for non-abelian loop Toda equations associated with general linear groups. Here we consider the untwisted case only and use the rational dressing method based upon appropriate block-matrix representation suggested by the initial $\bbZ$-gradation.
Three nonequivalent real forms of the complex twisted N=2 supersymmetric Toda chain hierarchy (solv-int/9907021) in real N=1 superspace are presented. It is demonstrated that they possess a global twisted N=2 supersymmetry. We discuss a new superfield basis in which the supersymmetry transformations are local. Furthermore, a representation of this hierarchy is given in terms of two twisted chiral N=2 superfields. The relations to the s-Toda hierarchy by H. Aratyn, E. Nissimov and S. Pacheva (solv-int/9801021) as well as to the modified and derivative NLS hierarchies are established.
Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with integrable Z-gradations is discussed.
We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitzéica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitzéica transformations, and discuss the group structure for these transformations.
We find new solutions, including soliton-like ones, for a special case of non-Abelian loop Toda equations associated with complex general linear groups. We use the method of rational dressing based on an appropriate block-matrix representation suggested by the Z-gradation under consideration. We present solutions in a form of a direct matrix generalization of the Hirota's soliton solution already well-known for the case of Abelian loop Toda systems.
On exact overlaps of integrable matrix product states: inhomogeneities, twists and dressing formulas
本文利用量子穿衣程序和扭曲 Yangian 表示论，推导出可积矩阵乘积态与 Bethe 本征态重叠的求和公式，并针对 SO(6) 自旋链填补了 D3-D5 畴壁系统重叠公式解析理解的最后空白。
We review some of the fundamental notions associated to the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux-Backlund transformation is introduced and employed in order to obtain soliton solutions for specific examp…
The concept of point-like "jump" defects is investigated in the context of affine Toda field theories. The Hamiltonian formulation is employed for the analysis of the problem. The issue is also addressed when integrable boundary conditions ruled by the classical twisted Yangian are present. In both periodic and boundary cases explicit expressions of conserved quantities as well as the relevant Lax pairs and sewing conditions are extracted. It is also observed that in the case of the twisted Yangian the bulk behavior is not affected by the presence of the boundaries.
We construct integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type …
Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff lattice"). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called "the Pfaff-Toda hierarchy"). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (diff…
Three inequivalent real forms of the complex N=4 supersymmetric Toda chain hierarchy (Nucl. Phys. B558 (1999) 545, solv-int/9907004) in the real N=2 superspace with one even and two odd real coordinates are presented. It is demonstrated that the first of them possesses a global N=4 supersymmetry, while the other two admit a twisted N=4 supersymmetry. A new superfield basis in which supersymmetry transformations are local is discussed and a manifest N=4 supersymmetric representation of the N=4 Toda chain in terms of a chiral and an anti-chiral N=4 superfield is constructed. Its relation to the …
On a class of exact solutions of the Ishimori equation
本研究针对在铁磁体理论中有重要应用的Ishimori方程，证明此前发现的二维Toda型格点为其装扮链，并利用该装扮链的可积约化得到一类本质全新的精确解。
The partition function of $\mathcal{N}=2$ super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual $Ω$-background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a ${\it non-autonomous}$ Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical so…
Generic classically integrable boundary conditions for the $A_{n}^{(1)}$ affine Toda field theories (ATFT) are investigated. The present analysis rests primarily on the underlying algebra, defined by the classical version of the reflection equation. We use as a prototype example the first non-trivial model of the hierarchy i.e. the $A_2^{(1)}$ ATFT, however our results may be generalized for any $A_{n}^{(1)}$ ($n>1$). We assume here two distinct types of boundary conditions called some times soliton preserving (SP), and soliton non-preserving (SNP) associated to two distinct algebras, i.e. the…
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