相关论文: Conformal weldings and Dispersionless Toda hierarc…
Let $\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a univalent function on the unit disc with $f(0)=0$, $g$ is a univalent function on the exterior of the unit disc with $g(\infty)=\infty$ and $f'(0)g'(\infty)=1$. In…
In this article, we classify the solutions of the dispersionless Toda hierarchy into degenerate and non-degenerate cases. We show that every non-degenerate solution is determined by a function $\mathcal{H}(z_1,z_2)$ of two variables. We…
We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy…
We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the…
In recent works [hep-th/9909147, hep-th/0005259] was found a wonderful correlation between integrable systems and meromorphic functions. They reduce a problem of effictivisation of Riemann theorem about conformal maps to calculation of a…
For a family of Poisson algebras, parametrized by by an integer number r, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th…
Under certain reality conditions, a general solution to the dispersionless Toda lattice hierarchy describes deformations of simply-connected plane domains with a smooth boundary. The solution depends on an arbitrary (real positive) function…
The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with $M+1$ marked points. These solutions are characterized by a Riemann-Hilbert problem…
The dispersionless KP and Toda hierarchies possess an underlying twistorial structure. A twistorial approach is partly implemented by the method of Riemann-Hilbert problem. This is however still short of clarifying geometric ingredients of…
We study the mixed Hessian of the dispersionless Toda $\tau$-function for the one-harmonic $s$-fold symmetric conformal map $f(w)=rw+aw^{1-s}$. This Hessian is the susceptibility matrix generated by the inverse conformal map. Our spectral…
Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of…
We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda…
We study finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures,…
We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the…
Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of…
Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is…
The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This…
Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on two-dimensional conformal field theories, Toda conformal field theories were introduced in the physics literature as a family of two-dimensional conformal field…
The integrable structure, recently revealed in some classical problems of the theory of functions in one complex variable, is discussed. Given a simply connected domain in the complex plane, bounded by a simple analytic curve, we consider…
We show that one-variable reductions of the Pfaff-Toda integrable hierarchy in the dispersionless limit are described by a system of coupled elliptic L\"owner (Komatu-Goluzin) equations.