Nonlinear matrix algebra and engineering applications. Part 1: Theory and linear form matrix

A matrix vector formalism is developed for systematizing the manipulation of sets of non-linear algebraic equations. In this formalism all manipulations are performed by multiplication with specially constructed transformation matrices. For many important classes of nonlinearities, algorithms based on this formalism are presented for rearranging a set of equations so that their solution may be obtained by numerically searching along a single variable. Theory developed proves that all solutions are obtained.     

The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

On rigged immersions

A self-contained account is given in an efficient formalism of rigged immersions of one manifold-with-connection in another, leading to the analogues of the Gauss, Codazzi and Ricci equations discovered by Schouten. The equations expressing their interdependence are then derived and it is shown that in general one of the two sets of “Codazzi” equations is a consequence of the other set and the Gauss and Ricci equations. The formalism is specialised to the Riemannian case, where it is shown that, for large codimension (specific limits being given), all butn components of the Codazzi equations are determined by the other equations. A local theorem on the existence of rigged immersions is proved.     

Effective matrix formalism for singularity analysis of differential equations and new intergrable system in nonlinear elasticity

We introduce a simple matrix formalism for Taylor series and generalized Laurent series that can be used for implementing the Taylor method for nonlinear ODEs and singularity analysis of differential equations. Advantages of this approach over conventional techniques are shown on model examples. Surprisingly, the same formalism can be used for proving C-integrability of a 3D model in nonlinear elasticity. An alternative proof is obtained by using similarity between the model in nonlinear elasticity and the classic Pohlmeier-Lund-Regge model from high energy physics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational formulations of differential equations and asymmetric fractional embedding

Variational formulations for classical dissipative equations, namely friction and diffusion equations, are given by means of fractional derivatives. In this way, the solutions of those equations are exactly the extremal of some fractional Lagrangian actions. The formalism used is a generalization of the fractional embedding developed by Cresson [Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys. 48 (2007) 033504], where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives. Moreover, this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle.     

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained.     

Homogenization of very rough interfaces for the micropolar elasticity theory

In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar isotropic micropolar elastic solids is investigated. The interface is assumed to oscillate between two parallel straight lines. The main aim is to derive homogenized equations in explicit form. These equations are obtained by the homogenization method along with the matrix formalism of the theory of micropolar elasticity. Since obtained homogenized equations are totally explicit, they are a powerful tool for solving various practical problems. As an example, the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type is investigated. The closed-form formulas for the reflection and transmission coefficients have been derived. Based on these formulas, some numerical examples are carried out to show the dependence of the reflection and transmission coefficients on the incident angle and the geometry parameter of the interface.     

Perturbation theory for the periodic Anderson model: II. Superconducting state

A new diagram technique, which has been developed for strongly correlated electron systems, is used to study the periodic Anderson model in the superconducting state. To treat both normal and anomalous Green's functions on an equal footing, we introduce an additional charge quantum number that distinguishes creation and annihilation operators. We derive the Dyson equations for the Green's functions of band and localized electrons in the presence of superconductivity. The equations obtained admit both singlet-type and triplet-type superconductivity. For singlet-type superconductivity, we establish the correspondence between these equations and the spinor Gor'kov-Nambu formalism. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 3, pp. 456–473, September, 1998.     

Nudelman interpolation and the band method

In previous work the authors developed a new addition of the band method based on a Grassmannian approach for solving a completion/extension problem in a general, abstract framework. This addition allows one to obtain a linear fractional representation of all solutions of the abstract completion problem from special extensions which are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). In this work we extend this framework to a somewhat more general setting and show how one can obtain formulas for the required special extensions from solutions of a system of linear equations. As an application we show how the formalism can be applied to the bitangential Nevanlinna-Pick interpolation problem, a case which, up to now, was not amenable to the band method.The first author was partially supported by National Science Foundation grant DMS-9500912.     

About fractional quantization and fractional variational principles

In this paper, a new method of finding the fractional Euler–Lagrange equations within Caputo derivative is proposed by making use of the fractional generalization of the classical Faá di Bruno formula. The fractional Euler–Lagrange and the fractional Hamilton equations are obtained within the 1 + 1 field formalism. One illustrative example is analyzed.     

Painlevé analysis, Lie symmetries and exact solutions for (2+1)-dimensional variable coefficients Broer-Kaup equations

A (2+1) dimensional Broer-Kaup system which is obtained from the constraints of the KP equation is of importance in mathematical physics field. In this paper, the Painlevé analysis of (2+1)-variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the Painlevé property. Similarity reductions of the VCBK equation to one-dimensional partial differential equations including Burger’s equation are investigated by the Lie classical method. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Threshold Phenomena in Intense Electromagnetic Fields

The influence of intense electromagnetic fields on the formation and decay of quasistationary states of different quantum systems is investigated based on exact solutions of quantum equations for charged particle motion. The method allows examining systems where a spontaneous decay may occur as well as phenomena that occur only under the action of the field. Different values of the total magnetic moment of the system are taken into account in this consideration. A consistent use of the analytic continuation method allows obtaining nonlinear equations that determine complex energies in an external field. The asymptotic expansions for real and imaginary energy values under the action of weak and strong electromagnetic fields are investigated. The developed approach allows establishing the characteristic values for the length parameters that determine the formation of the processes in superstrong fields. We note that a significant decrease of distances in strong fields may lead to effects with a new characteristic length scale, characterizing a modified quantum electrodynamics (QED) formalism, namely, the QED with the fundamental mass formalism.     

Exact Solutions of Some Nonlinear Evolution Equations

A different approach to finding solutions of certain diffusive-dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, which is carried to all terms, followed by a summation of the resulting infinite series. Sometimes this is done directly and other times in terms of inverses of operators in an appropriate space. We first illustrate the method with Burgers's and Thomas's equations, and show how it quickly leads to the Cole-Hopf and Thomas transformations which linearize these equations. The method is described in detail with the Korteweg-de Vries equation and then applied to the modified KdV, sine-Gordon, nonlinear (cubic) Schrödinger, complex modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained, and new expressions for some of them follow. More generally, the Mar?enko integral equations, together with the inverse problem that originates them, follow naturally from the approach. A method for modifying known solutions (in a way different from the known Backlund transformations) is also developed. Thus, for example, formulas for the interaction of solitons with an arbitrary given solution are obtained. Other equations tractable by this approach are presented. These include the vector-valued cubic Schrödinger equation and a two-dimensional nonlinear Schrödinger equation. Higher-order and matrix-valued equations with nonscalar dispersion functions are also included.     

An initial value theory for linear causal boundary value problems

A new formalism in the theory of linear boundary value problems involving causal functional differential equations is presented. The approach depends on the construction of a differentiable family of boundary problems into which the original boundary value problem is imbedded. The formalism then generates an initial value problem which is equivalent to the family of imbedded problems. An important aspect of the method is that the equations in the initial value algorithm are ordinary differential equations rather than functional differential equations, although nonlinear and of higher dimension. Applications of the theory to differential-delay and difference equations are given.     

Method of anomalous Green's functions: Antiferromagnetism in the Hubbard model on a triangular lattice

A method is proposed for describing antiferromagnetic and ferromagnetic states on a triangular lattice in the formalism of anomalous temperature-dependent Green's functions, for which equations of Dyson-Gor'kov type are formulated. These equations are solved in the Hartree approximation, and self-consistency equations are obtained for the order parameters. Finally, the connection between the considered theory and experiment is discussed.deceasedSt. Petersburg Branch of the V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 294–303, November, 1994.     

Benjamin-Bona-Mahony (BBM) equation with variable coefficients: Similarity reductions and Painlevé analysis

The Lie-group formalism is applied to investigate the symmetries of the Benjamin-Bona-Mahony (BBM) equation with variable coefficients. We derive the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Ray Method Expansions for Surface and Internal Waves in Inhomogeneous Oceans of Variable Depth

A ray method formalism is developed for the analysis of surface and internal waves in an inhomogeneous ocean of variable depth. In this method, we deduce from the governing system of equations a system of first order ordinary differential equations, for the group lines (rays of the ray method) and the propagation of phase and amplitude on them. The dispersion relation for these waves arises as an eigen-condition on an eigen-value problem involving an ordinary differential equation in the depth variable. The deduced equation for amplitude propagation has the interpretation of a statement of conservation of action.     

Linear Volterra Integral Equations

The Kurzweil-Henstock integral formalism is applied to establish the existence of solutions to the linear integral equations of Volterra-typewhere the functions are Banach-space valued. Special theorems on existence of solutions concerning the Lebesgu3 integral setting are obtained. These sharpen earlier results.