Novel curved solitons of integrable (2 +1) dimensional KMN equation

On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with ano…

A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

A new integrable generalization to the 2D sphere $S^2$ and to the hyperbolic space $H^2$ of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian $H_κ$, we will make use of a group theoretical approach in which the curvature $κ$ of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. It turns out …

Matrix extension of multidimensional dispersionless integrable hierarchies

We consistently develop a recently proposed scheme of matrix extension of dispersionless integrable systems for the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax pairs, Lax-Sato equations, matrix equations on the background of vector fields and the dressing scheme. Reductions, construction of solutions and connections to geometry are discussed. We consider separately a case of Abelian extension, for which the Riemann-Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of Penrose for…

一个（2+1）维可积自旋模型：几何和规范等价对应物、孤子和局域相干结构

A (2+1) dimensional integrable spin model: Geometrical and gauge equivalent counterpart, solitons and localized coherent structures

研究了一类非等谱（2+1）维可积自旋方程.结果表明，它的几何和规范等效方程是由Zakharov提出并由Strachan最近研究的（2+1）维非线性薛定谔方程。使用广田双线性化的形式，线和弯曲孤子的解决方案。利用双线性化方程解的一定自由度（任意性），得到了势函数的指数定域dromion类解，并构造了冲击波型和代数定域性质的破缺孤子解（自旋变量）.

Universal quantum Hawking evaporation of integrable two-dimensional solitons

We show that any soliton solution of an arbitrary two-dimensional integrable equation has the potential to eventually evaporate and emit the exact analogue of Hawking radiation from black holes. From the AKNS matrix formulation of integrability, we show that it is possible to associate a real spacetime metric tensor which defines a curved surface, perceived by the classical and quantum fluctuations propagating on the soliton. By defining proper scalar invariants of the associated Riemannian geometry, and introducing the conformal anomaly, we are able to determine the Hawking temperatures and e…

关于最简单的（2+1）维可积自旋系统及其等价的非线性薛定谔方程

On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schrödinger equations

利用运动空间曲线形式，建立了（2+1）维自旋方程（M-I方程）与（2+1）维非线性薛定谔方程（NLSE）之间的几何等价性和规范等价性.得到了一组相容的三个线性方程，并讨论了运动积分。通过球极投影，将M-I方程双线性化，得到了线孤子、弯曲孤子、破缺孤子、诱导孤子和畴壁型解。（2+1）维NLSE的破缺孤子解也得到了报道。上述自旋方程的推广进行了讨论。

The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

An integrable generalization on the two-dimensional sphere S^2 and the hyperbolic plane H^2 of the Euclidean anisotropic oscillator Hamiltonian with "centrifugal" terms given by $H=1/2(p_1^2+p_2^2)+ δq_1^2+(δ+ Ω)q_2^2 +\frac{λ_1}{q_1^2}+\frac{λ_2}{q_2^2}$ is presented. The resulting generalized Hamiltonian H_κ depends explicitly on the constant Gaussian curvature κ of the underlying space, in such a way that all the results here presented hold simultaneously for S^2 (κ>0), H^2 (κ<0) and E^2 (κ=0). Moreover, H_κ is explicitly shown to be integrable for any values of the parameters δ, Ω, λ_1 and…

可积辅助场σ模型与类$T\bar{T}$形变的度规方法

Integrable auxiliary field sigma model and the metric approach to $T\bar{T}$-like deformations

该研究提出一类保持可积性的二维度规形变，通过主作用量关联不同辅助场产生的形变，证明其保持经典Lax可积性，并确定$T\bar{T}$是唯一可解释为场依赖坐标变换的类$T\bar{T}$形变。

扭量理论与微分方程

Twistor Theory and Differential Equations

这是扭量理论作为解决非线性微分方程的几何工具的一个基本的和独立的审查。孤子方程如KdV、Tzitzeica、可积手征模型、BPS、Sine-Gordon等的解都来自于$T\CP^1$上的全纯向量丛。对孤子方程的无色散类似物，如2+1维的无色散KP或$SU(\infty)$ Toda系统，提供了一个不同的框架。他们的解决方案对应的变形（部分）$T\CP^1$，并最终爱因斯坦-外尔弯曲几何推广平闵可夫斯基空间。一些练习包括和必要的事实有关向量丛超过黎曼领域的总结在附录中。