具有离散Laplacian的周期ILW方程

Self-dual form of Ruijsenaars-Schneider models and ILW equation with discrete Laplacian

We discuss a self-dual form or the Bäcklund transformations for the continuous (in time variable) ${\rm gl}_N$ Ruijsenaars-Schneider model. It is based on the first order equations in $N+M$ complex variables which include $N$ positions of particles and $M$ dual variables. The latter satisfy equations of motion of the ${\rm gl}_M$ Ruijsenaars-Schneider model. In the elliptic case it holds $M=N$ while for the rational and trigonometric models $M$ is not necessarily equal to $N$. Our consideration is similar to the previously obtained results for the Calogero-Moser models which are recovered in t…

Non-chiral Intermediate Long Wave equation and inter-edge effects in narrow quantum Hall systems

We present a non-chiral version of the Intermediate Long Wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account inter-edge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws for this new equation. We also present a periodic non-chiral ILW equation, together with its soliton solutions governed by the elliptic CMS model.

六维超对称规范理论、瞬子模空间的量子上同调和gl（N）量子中长波流体力学

Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics

我们证明了在C^2 × S^2上具有八个超荷的U（N）六维规范理论的精确配分函数提供了称为gl（N）周期中长波（ILW）的流体力学型可积系统的量子化。我们把这个系统刻画为椭圆型Calogero-Moser可积系统的流体动力极限。我们从S^2上的有效规范线性西格玛模型出发，以ADHM瞬子模空间为目标空间，计算Bethe方程，其镜像计算gl（N）ILW的Yang-Yang函数。量子哈密顿量由六维规范理论的定域手征环观测量给出。作为特殊情况，它们提供了gl（N）Benjamin-Ono和Korteweg-de Vries量子哈密顿量。在四维极限下，我们用Heisenberg + W_N代数的守恒荷确定了局部手征环观测量，从而给出了AGT对应的规范理论证明。

具有离散Laplacian的周期Benjamin-Ono方程的一些特解

On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian

我们研究了一个周期版本的Benjamin-Ono（BO）方程与离散拉普拉斯算子。我们找到了这个方程的一些特解，并计算了与这些解对应的前两个运动积分$I_1$和$I_2$的值。发现它们与Macdonald $q$差分算子的谱有很强的相似性。为了更好地理解这些经典和量子可积系统之间的联系，我们考虑了特殊的退化情况下相应的$q=0$更详细。也就是说，我们给出了这种退化周期BO的通解，得到了表示所有运动积分$I_n$（$n=1,2,...$）的显式公式，并成功地将其与Macdonald算子在极限$q\to 0$（Macdonald多项式趋于Hall-Littlewood多项式的极限）中的特征值相鉴别.

孤立子理论中的Laplacian增长和Whitham方程

Laplacian Growth and Whitham Equations of Soliton Theory

本文证明了零表面张力下多连通区域的Laplacian增长问题（Hele-Shaw问题）等价于孤立子理论中已知的Whitham方程的一个可积系统。Whitham方程描述了可积非线性微分方程族的慢调制周期解。通过这种联系，拉普拉斯增长被理解为黎曼曲面的模空间中的流动。

Periodic waves in the fractional modified Korteweg--de Vries equation

Periodic waves in the modified Korteweg-de Vries (mKdV) equation are revisited in the setting of the fractional Laplacian. Two families of solutions in the local case are given by the sign-definite dnoidal and sign-indefinite cnoidal solutions. Both solutions can be characterized in the general fractional case as global minimizers of the quadratic part of the energy functional subject to the fixed $L^4$ norm: the sign-definite (sign-indefinite) solutions are obtained in the subspace of even (odd) functions. Morse index is computed for both solutions and the spectral stability criterion is deri…