可精确求解与可积系统
It is shown that the $N$-dark soliton solutions of the integrable discrete nonlinear Schr\"odinger (IDNLS) equation are given in terms of the Casorati determinant. The conditions for reduction, complex conjugacy and regularity for the…
A transforation between a hierarchy of integrable equations arising from the standard $R$-matrix construction on the algebra of differential operators and a hierarchy of integrable equations arising from a deformation of the standard…
We discuss computer implementation of the known algorithm of finding separation coordinates for the special class of orthogonal separable systems called L-systems or Benenti systems.
We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Lax representation…
New Cellular Automata associated with the Schroedinger discrete spectral problem are derived. These Cellular Automata possess an infinite (countable) set of constants of motion.
First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation and next we perform a reduction of the dKP equation to the discrete 1D Toda equation. This gives a method of…
We discuss a method of solving $n^{th}$ order scalar ordinary differential equations by extending the ideas based on the Prelle-Singer (PS) procedure for second order ordinary differential equations. We also introduce a novel way of…
We consider the cubic and quartic He'non-Heiles Hamiltonians with additional inverse square terms, which pass the Painleve' test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the…
We consider a family of integro-differential equations depending upon a parameter $b$ as well as a symmetric integral kernel $g(x)$. When $b=2$ and $g$ is the peakon kernel (i.e. $g(x)=\exp(-|x|)$ up to rescaling) the dispersionless…
he Singular Manifold Method is presented as an excellent tool to study a 2+1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1+1 reductions of the same equation. Nevertheless these…
We consider a long--range homogeneous chain where the local variables are the generators of the direct sum of $N$ $\mathfrak{e}(3)$ interacting Lagrange tops. We call this classical integrable model rational ``Lagrange chain'' showing how…
We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to…
Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several O(N)-invariant classes of hyperbolic equations $u_{tx} =f(u,u_t,u_x)$ for an $N$-component vector $u(t,x)$ are considered. In each class we…
The phenomenon of steady streaming, or acoustic streaming, is an important physical phenomenon studied extensively in the literature. Its mathematical formulation involves the Navier-Stokes equations, thus due to the complexity of these…
Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered. Addition formula for the $\tau$-function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry…
A soliton cellular automaton on a one dimensional semi-infinite lattice with a reflecting end is presented. It extends a box-ball system on an infinite lattice associated with the crystal base of U_q(sl_n). A commuting family of time…
In this contribution, we give an integral representation of the wave functions of the quantum N-particle Toda chain with boundary interaction. In the case of the Toda chain with one-boundary interaction, we obtain the wave function by an…
We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a $\tau$-function. We investigate the potential using the non-local $\bar\partial$ dressing…
We perform a In\"on\"u--Wigner contraction on Gaudin models, showing how the integrability property is preserved by this algebraic procedure. Starting from Gaudin models we obtain new integrable chains, that we call Lagrange chains,…
In this review we explain interrelations between the Elliptic Calogero-Moser model, integrable Elliptic Euler-Arnold top, monodromy preserving equations and the Knizhnik-Zamolodchikov-Bernard equation on a torus.