可精确求解与可积系统
As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse…
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution $u$. Then we construct a differential (respectively, difference)…
A one-to-one correspondence between triple Jordan systems and integrable multi-component models of the modified Korteveg--de Vries type is established.
We demonstrate the systematic derivation of a class of discretizations of nonlinear Schr{\"o}dinger (NLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic condition. We…
Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for $(1+1)$-dimensional PDEs can be interpreted as ZCRs. In [arXiv:1303.3575], for any $(1+1)$-dimensional scalar…
A chain of transformations is found which relates one new integrable case of the generalized short pulse equation of Hone, Novikov and Wang [arXiv:1612.02481] with the sine-Gordon equation.
In this paper we ivestigate St\"ackel transforms between different classes of parameter-dependent St\"ackel separable systems of the same dimension. We show that the set of all St\"ackel systems of the same dimension splits to equivalence…
This paper connects the quadrirational Yang-Baxter maps, which are two-dimensional integrable discrete systems of KdV type, and the elliptic Cremona system, which is a higher analogue of discrete Painlev\'e equations associated with…
We obtain the well-known discrete modified Boussinesq equation in two-component form as well as its Lax pair in $3\times3$ matrix form through a 3-periodic reduction technique on the Hirota-Miwa equation and its Lax pair. We describe how…
We consider the issue of deriving superintegrable systems with position dependent mass (PDM) in two dimensions from certain known superintegrable systems using the recently introduced method of master symmetries and complex factorization by…
We study several variants of q-Garnier system corresponding to various directions of discrete time evolutions. We also investigate a relation between the $q$-Garnier system and Suzuki's higher order $q$-Painlev/'e system by using a duality…
The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The use of the exotic cohomology of the symmetry algebras opens a way to formulate such…
We establish a new connection between the theory of totally positive Grassmannians and the theory of $\mathtt M$-curves using the finite--gap theory for solitons of the KP equation. Here and in the following KP equation denotes the…
In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear…
We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear…
We employ the Ablowitz-Ladik system as an illustrative example in order to demonstrate how to analyze initial-boundary value problems for integrable nonlinear differential-difference equations via the unified transform (Fokas method). In…
The aim of this paper is to introduce the Inverse Scattering Method for later studies of some problems in nonlinear dynamics, and describe the kink solution of the Sine Gordon Equation using the Inverse Scattering Method as a methodological…
We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By…
We consider an auxiliary spectral problem originally introduced by Gerdjikov, Mikhailov and Valchev (GMV system) and its modification called pseudo-Hermitian reduction which is extensively studied here for the first time. We describe the…
We consider the stationary sine-Gordon equation on metric graphs with simple topologies. The vertex boundary conditions are provided by flux conservation and matching of derivatives at the star graph vertex. Exact analytical solutions are…