可精确求解与可积系统

We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties…

The problem of searching boundary value problems for soliton equations consistent with the integrability property is discussed. A method of describing integrals of motion for the integrable initial boundary value problems for the…

A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of…

We apply Cartan's method of equivalence to find a B\"acklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.

Recently we proposed a generic construction of the additional integrals of motion for the St\"ackel systems applying addition theorems to the angle variables. In this note we show some trivial examples associated with angle variables for…

We develop the Inverse Scattering Transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an $\mathfrak{sl}(3)$ Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic…

The complete integrability of the Ostrovsky-Vakhnenko equation is studied by means of symplectic gradient-holonomic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, Lax type representation and related…

We give integrable quad equations which are multi-quadratic (degree-two) counterparts of the well-known multi-affine (degree-one) equations classified by Adler, Bobenko and Suris (ABS). These multi-quadratic equations define multi-valued…

We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which…

We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on $E^2$…

We use the Fokas method to analyze the derivative nonlinear Schr\"odinger (DNLS) equation $iq_t(x,t)=-q_{xx}(x,t)+(r q^2)_x$ on the interval $[0,L]$. Assuming that the solution $q(x,t)$ exists, we show that it can be represented in terms of…

The semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials is studied in a full scaling neighborhood D of the point of gradient catastrophe (x_0,t_0). This…

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Li\'enard type oscillators exhibit this interesting property. We show that a…

We study amplitudes of five-wave interactions for evolution Hamiltonian equations differ from the KdV equation by the form of dispersion law. We find that five-wave amplitude is canceled for all three known equations (KdV, Benjamin-Ono and…

By introducing Lenard recursion equations, we derive a general coupled nonlinear Sch$\mathrm{\ddot{o}}$dinger (CNLS) hierarchy associated with well-known Manakov system and Sasa-Satsuma system. Based on the characteristic polynomial of Lax…

We report triangular auto-B\"acklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup-Kupershmidt equation derived by E. G. Reyes in his paper titled…

We define a new grading, that we call the "level grading", on the algebra of polynomials generated by the derivatives $u_{k+i}=\partial^{k+i}u/\partial x^{k+i}$ over the ring $K^{(k)}$ of $C^{\infty}$ functions of $u,u_1,...,u_k$. This…

An approach based on the spectral and Lie - algebraic techniques for constructing vertex operator representation for solutions to a Riemann type Gurevicz-Zybin hydrodynamical hierarchy is devised. A functional representation generating an…

The algebraic structure of the integrable mixed mKdV/sinh-Gordon model is discussed and \textit{}extended to the AKNS/Lund-Regge model and to its corresponding supersymmetric versions. The integrability of the models is guaranteed from the…

We start with a Riemann-Hilbert Problems (RHP) with canonical normalization whose sewing functions depends on several additional variables. Using Zakharov-Shabat theorem we are able to construct a family of ordinary differential operators…