Theoretical and Mathematical Physics最新文献

We propose a sequence of Lax pairs whose compatibility conditions are three-component integrable nonlinear equations. The first equations of this hierarchy are the three-component Kaup–Newell, Chen–Lee–Liu, and Gerdjikov–Ivanov equations. The type of equation depends on an additional parameter (alpha). The proposed form of the three-component Kaup–Newell equation is slightly different from the classical one. We show that the evolution of the components of the simplest nontrivial solutions of these equations is completely determined by the evolution of the length of the solution vector and additional numerical parameters.

We explore integrable equations that involve involution points, along with the solution phenomena for Cauchy problems associated with nonlocal differential equations. By applying group reductions to classical Lax pairs, we generate nonlocal integrable equations. Soliton solutions of these models are derived using binary Darboux transformations or reflectionless Riemann–Hilbert problems in the nonlocal context. Further discussion on the well-posedness of nonlocal differential equations is also presented.

We consider some special systems of integro-differential equations, the so-called integral networks of nonlinear oscillators. These networks are obtained from finite-dimensional fully connected networks when the number of interacting oscillators tends to infinity. We study both general properties of the introduced class of equations and the characteristic features of the dynamics of integral networks. Namely, we establish the fundamental possibility of the existence of so-called periodic regimes of multicluster synchronization in these networks. For any such regime, the set of oscillators decomposes into (r), (rge 2), nonintersecting classes. Within these classes, complete synchronization of oscillations is observed, and every two oscillators from different classes oscillate asynchronously. We also establish the realizability of the phenomenon of continuum buffering, that is, of the existence under certain conditions of a continuum family of isolated attractors.

We consider a three-point initial boundary value problem for a nonlinear functional partial differential equation with an infinite (integral) delay in the argument. The boundary conditions contain a delay in the argument and the highest derivative with respect to time. The initial boundary value problem is a mathematical model of the dynamics of a distributed rotating ideal shaft (rotor) of constant cross section with an ideal rigid circular disk mounted on the shaft. The axes of the shaft and disk coincide, the ends of the shaft rest on bearings. It is assumed that the shaft material obeys a nonlinear rheological model of a hereditarily elastic body. A definition of a solution of the initial boundary value problem is given based on the variational principle. Function spaces for the initial conditions and solutions are introduced, the phase space of the initial boundary value problem is defined. The existence theorem is proved for a solution, as is its uniqueness and continuous dependence on the initial conditions and parameters of the initial boundary value problem in the norm of the phase space. Thus, we demonstrate the well-posedness of the considered initial boundary value problem.

We consider a periodic boundary-value problem for a nonlinear partial differential equation containing terms with a deviating spatial argument. The functional-differential equation under consideration was previously proposed as a model for describing the process of relief formation on a surface of semiconductors under ionic bombardment. We show that the boundary-value problem under consideration can have an asymptotically large number of two-dimensional invariant manifolds formed by solutions that have the structure of traveling periodic waves. We also show that these invariant manifolds are typically saddle ones, and the number of those that are local attractors does not exceed two. We obtain asymptotic formulas for solutions belonging to a given invariant manifolds. These mathematical results partially explain the complexity of dynamics of pattern formation on the surface of semiconductors.

We address the existence of solitons and periodic traveling-wave solutions in a saturable discrete NLS (dNLS) equation with next-nearest-neighbor interactions. Calculus of variations and Nehari manifolds are employed to establish the existence of discrete solitons. We prove the existence of periodic traveling waves studying the mixed-type functional differential equations using Palais–Smale conditions and variational methods.

We study self-distributive algebraic structures: algebras, bialgebras, additional structures on them, relations of these structures with Hopf algebras, Lie algebras, Leibnitz algebras, etc. The basic example of such structures is given by rack and quandle bialgebras. But we go further to the general coassociative comultiplication. The principal motivation for this work is the development of linear algebra related to the notion of a quandle in analogy with the ubiquitous role of group algebras in the category of groups with possible applications to the theory of knot invariants. We describe self-distributive algebras and show that some quandle algebras and some Novikov algebras are self-distributive. We also give a full classification of counital self-distributive bialgebras in dimension (2) over (mathbb{C}).

A space–time trigger model of tumor growth is considered depending on the concentration of hydrogen ions, oxygen, and tumor cell density at the initial stage of the tumor spheroid development. A system of parabolic equations with a piecewise linear right-hand side of modular type is used to solve this problem. Numerical implementation is carried out in a three-dimensional domain shaped as a cube with an edge of 0.1 mm on a uniform grid using the method of lines, the Rosenbrock scheme, and the factorization method in the spatial coordinates. The presented model quite well describes the dynamics of variation in the spheroid area at the initial stage of the tumor development depending on time. A distinctive feature of the model is that it reflects both the process of tumor growth into the external environment and the formation of a necrotic core at the tumor center. Based on the presented system of equations, it is possible to design a model that takes both the heterogeneity of the environment and more complex mechanisms of tumorigenesis into account.

The elliptic lattice KdV system, discovered in 2003, is an extension of the lattice potential KdV equation associated with an elliptic curve. This is a rather complicated three-component system on the quad lattice, which contains the moduli of the elliptic curve as parameters. In this paper, we investigate this system further and, among other results, derive a two-component multiquartic form of the system on the quad lattice. Furthermore, we construct an elliptic Yang–Baxter map and study the associated continuous and semidiscrete systems. In particular, we derive the so-called “generating PDE” for this system, comprising a six-component system of second-order PDEs, which can be considered to constitute an elliptic extension of the Ernst equations of General Relativity.

We present a method for constructing hierarchies of solutions of (n)-simplex equations by varying the spectral parameter in their Lax representation. We use this method to derive new solutions of the set-theoretic (2)- and (3)-simplex equations that are related to the Adler map and nonlinear Schrödinger (NLS) type equations. Moreover, we prove that some of the derived Yang–Baxter maps are completely integrable.