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In this paper, we study a system of nonlinear equations on a square graph related to the affine algebra A(1)1 . This system is the simplest representative of the class of discrete systems corresponding to affine Lie algebras. We find the Lax representation and construct hierarchies of higher symmetries. In neighborhoods of singular points ⋋ = 0 and ⋋ = ∞, we construct formal asymptotic expansions of eigenfunctions of the Lax pair and, based on these expansions, find series of local conservation laws for the system considered.

The problem of anthropogenic impact of the Tudakul reservoir has not been spared. In the first years of the reservoir’s existence, its ichthyofauna was formed by the fish of the Zarafshan River, which were represented by both rheophiles and limnophiles, and initially it consisted of 11 aboriginal species and introduced fish. This paper gives a brief physical-geographical, hydrological, hydrochemical, hydrobiological characteristics of the Tudakul reservoir. An assessment of the ecological state of the reservoir, ichthyofauna and fish catch in it over the past decade is given. Recommendations for the rational use of the Tudakul reservoir are given.

The study was carried out in 2014-2018 at the Tudakul reservoir, Uzbekistan. Samples included 278 European bream (Abramis brama), and 83 Amur white bream (Parabramis pekinensis). The age, total length and weight of the fish varied from 1 to 5 years, from 14 to 50 cm, from 30 to 1,405 g for the European bream and from 1 to 6 years, from 24 to 53.5 cm and 105 – 2,138 g for the white Amur bream and European bream, respectively. The relationship between total length (TL) and weight (W) is described by the equation W = 0.005 * TL3.2555 (r = 0.98) for European bream and W = 0.005 * TL3.201 (r = 0.96) for white Amur bream. Recovered average growth rate of European bream: TL1 - 13.5 cm; TL2 - 28.1 cm, TL3 - 37.95 cm, TL4 - 42.6 cm, TL5 - 48.6 cm.The average height of the Amur white bream: TL1 - 12.6 cm, TL2 - 23.3 cm, TL3 - 31.5 cm, TL4 - 37.9 cm, TL5 - 41.7 cm, TL6 - 44.2 cm. Lee’s phenomenon did not appear. All yearling fish were juvenile. In 2-year-old fish of both sexes of both species, the gonads were at stage II. Males and females of both species reached their first maturity at 3-4 years of age, when the total body length of the European bream was 30–32 cm, and the Amur white bream was 34–35 cm.

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L1−1 L2, where the linear differential operators L1 and L2 are chosen such that the ordinary differential equation (L2 −λL1)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L2, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L1 $$\tilde U$$ = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution $$\tilde U$$ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.

The dynamics of a pulsed signal propagating in an underground pipeline with a damaged section is investigated. Pressure waves propagating in a linear pipeline filled with a fluid are considered. Due to viscous friction and thermal conductivity, dissipation is taken into account in a thin liquid or gas layer near the wall. It is assumed that the damaged pipeline section is a reflecting surface. Conditions on this surfaces are obtained under the assumption that the rate of liquid leakage during wave propagation through the damaged section is determined by the permeability of the soil. It is shown that in weakly permeable soils, signals do not provide full information on damage, and in well-permeable soils, only short (high-frequency) pulsed signals can be used to detect damage.

A study is made of the evolution of pressure perturbations that propagate in a tubular channel filled with liquid or gas and having a damaged portion in the form of fairly large slots when their hydraulic resistance is minor. For this purpose, the authors have adopted a mathematical model taking account of viscous friction and of the transfer of heat in a thin layer of the liquid (or the gas) near the wall. Based on this model, dispersion relations have been obtained for the processes of propagation and reflection of harmonic waves from the damaged portion. An analysis has been made of the dependences of the phase velocity, the attenuation factor, and also the coefficients of reflection and transmission on the circular frequency, the physical properties of the liquid or the gas, and also the size of damage. The authors have presented results illustrating the dynamics of pulsed signals as a function of the distance to the damaged portion from the inlet cross section of the channel, the damage scale, and of the form of the medium (liquid or a gas) in the channel.

The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward the following hypothesis. Any integrable equation of the type under consideration admits an infinite sequence of finite-field Darboux-integrable reductions. The property of Darboux integrability of a finite-field system is formalized as finite-dimensionality condition of its characteristic Lie-Rinehart algebras. That allows one to derive effective integrability conditions in the form of differential equations on the right hand side of the equation under study. To test the hypothesis, we use known integrable equations from the class under consideration. In this article, we show that all known examples do have this property., Comment: 24 pages

We continue the previously started study of the development of a direct method for constructing the Lax pair for a given integrable equation. This approach does not require any addition assumptions about the properties of the equation. As one equation of the Lax pair, we take the linearization of the considered nonlinear equation, and the second equation of the pair is related to its generalized invariant manifold. The problem of seeking the second equation reduces to simple but rather cumbersome calculations and, as examples show, is effectively solvable. It is remarkable that the second equation of this pair allows easily finding a recursion operator describing the hierarchy of higher symmetries of the equation. At first glance, the Lax pairs thus obtained differ from usual ones in having a higher order or a higher matrix dimensionality. We show with examples that they reduce to the usual pairs by reducing their order. As an example, we consider an integrable double discrete system of exponential type and its higher symmetry for which we give the Lax pair and construct the conservation laws.

A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier it has been observed that this kind objects provide an effective tool for evaluating the recursion operators and Lax pairs. In this article we show with an example of the Volterra chain that the generalized invariant manifold can be used for constructing exact particular solutions as well. To this end we first find an invariant manifold depending on two constant parameters. Then we assume that ordinary difference equation defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive ordinary difference and differential equations, for the roots of the polynomials. Efficiency of the method is approved by some illustrative examples.

Systems of discrete equations on a quadrilateral graph related to the series of the affine Lie algebras are studied. The systems are derived from the Hirota-Miwa equation by imposing boundary condi...

It is widely known that the recursion operator is a very important component of integrability. It allows one to describe in a compact form both hierarchies of the generalized symmetries and infinite series of the local conservation laws. In the literature, we can find several methods for constructing recursion operators, some of them use the Hamiltonian approach and the others are based on the Lax representation of the equation. In the present article we discuss on an alternative method, suggested in \cite{HabKhaTMP18}, which is connected only with the first several generalized symmetries of the given equation. Efficiency of the method is illustrated with the examples of KdV, Krichever-Novikov and Kaup-Kupershmidt equations and discrete models., Comment: 22 pages

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 effectively be found. Actually it is defined by a quadratic form. As a result we get a nonlinear Lax pair consisting of the linearized equation and the invariant manifold. Our second step consists of finding an appropriate change of the variables to linearize the found nonlinear Lax pair. The desired change of the variables is again defined by a quadratic form. The method is illustrated by the well-known KdV equation and the modified Volterra chain. New Lax pairs are found. The formal asymptotic expansions for their eigenfunctions are constructed around the singular values of the spectral parameter. By applying the method of the formal diagonalization to these Lax pairs the infinite series of the local conservation laws are obtained for the corresponding nonlinear models., Comment: 16 pages

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 immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.