Related papers: On some linear equations associated with dispersio…

A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold to the linearization of the given equation. Examples show that such invariant manifold does exist and can…

We present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a Lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect…

A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the…

A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the Calogero-Moser and Toda systems, as…

The new examples are found of the constraints which link the 1+2-dimensional and multifield integrable equations and lattices. The vector and matrix generalizations of the Nonlinear Schr\"odinger equation and the Ablowitz-Ladik lattice are…

We present a novel general framework to deal with forward and backward components of the electromagnetic field in axially-invariant nonlinear optical systems, which include those having any type of linear or nonlinear transverse…

We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of $2\times 2$ matrices for the whole hierarchy, we construct the…

The dispersionless integrable system we consider here was introduced to the literature rather recently, it is connected with the general local form of self-dual conformal structure (SDCS) for the signature (2,2). In integrability framework…

This paper deals with the state estimation of non-linear and non-Gaussian systems with an emphasis on the numerical solution to the Bayesian recursive relations. In particular, this paper builds upon the Lagrangian grid-based filter (GbF)…

We consider systems of diffusion equations that have considerable interest in Soil Science and Mathematical Biology and focus upon the problem of finding those forms of this class that can be linearized. In particular we use the equivalence…

From a specific series of exchange conditions for a one-parameter Hamiltonian vector field, we establish an integrable hierarchy using Lax pairs derived from the dispersionless partial differential equation. An exterior differential form of…

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is…

The system of two nonlinear coupled oscillators is studied. As partial case this system of equation is reduced to the Duffing oscillator which has many applications for describing physical processes. It is well known that the inverse…

Many finite dimensional integrable systems qre expressed with the help of the Lax equation which highlights a spectral parameter and therefore a spectral curve. These spectral curves are the starting point of an algebro-geometric…

A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…

Matrix completion aims to predict missing elements in a partially observed data matrix which in typical applications, such as collaborative filtering, is large and extremely sparsely observed. A standard solution is matrix factorization,…

We propose a Lax equation for the non-linear sigma model which leads directly to the conserved local charges of the system. We show that the system has two infinite sets of such conserved charges following from the Lax equation, much like…

We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found…

A solution of linear systems of equations Ax=b and Ax=0 is a vital part of many computational packages. This paper presents a novel formulation based on the projective extension of the Euclidean space using the outer product (extended…

Field equations for generalized principle models with nonconstant metric are derived and ansatz for their Lax pairs is given. Equations that define the Lax pairs are solved for the simplest solvable group. The solution is dependent on one…