ALBERT

Description: In this paper we investigate the existenceof the positive solutions for the followingnonlinear Schr\"odinger equation$$-\triangle u+V(x)u=K( x)|u|^{p-2}u\ \mbox{in}\ \mathbb{R}^N$$where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as$|x|\rightarrow\infty$ with $0

Description: In this paper, the approximate solutions of second-order linear differential equation with fuzzy boundary conditions in which coefficient functions maintain the sign, is investigated. The fuzzy linear bound value problem is converted to a crisp function system of linear equations by the undetermined fuzzy coefficients method. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method.

Description: This paper is concerned with a general condition for the solvability ofintegral geometry problems along the plane curves of given curvatures. Astwo important results, the solvabilities of integral geometry problems alongthe family of circles with fixed radius and along the family of circles ofvarying radius centered on a fixed circle are given. By using some extensionof the class of unknown functions, the proofs are based on the solvabilitiesof equivalent inverse problems for transport-like equation.

Description: Biography of Professor Hari M. Srivastava

Description: In this paper, we focus on the following second-order multi-point integral boundary-value problem$$u''(t)+a(t)f(u(t))=0, 0

Description: This work is focused on a system of boundary value problemswhose solutions represent the equilibria of a bridge suspended by continuously distributed cables and supported by M intermediate piers. The road bed is modeled as the junction of N=M+1 extensible elastic beams which are clamped each other and pinned at their ends to each pier. The suspending cables are modeled as one-sided springs with stiffness k. Stationary solutions of these doubly nonlinear problems are explicitly and analytically derived for arbitrary k and a general axial load p applied at the ends of the bridge. In particular, we scrutinize the occurrence of buckled solutions in connection with the length of each sub-span of the bridge.

Description: Using the M$\ddot{o}$nch fixed point theorem and progressive estimation method, we obtain the existence results of mild solutions for impulsive partial neutral second order integro-differential evolution equations with infinite delay in Banach spaces. The compactness condition of the impulsive term $I^j_i(\cdot)$, the Lipschitz condition of $I^j_i(\cdot)$ and $g_j(t,\cdot) \ (j=1,2)$, some restrictive conditions on a priori estimation and noncompactness measure estimation have been deleted. Our conditions are simple and our results essentially improve and extend some known results. As an application, an example is provided to illustrate the obtained results.

Description: In this paper, we consider a discontinuous Sturm-Liouville operator with parameter-dependent boundary conditions and two interior discontinuities. We obtain eigenvalues and eigenfunctions together with their asymptotic approximate formulas. Then, we give some uniqueness theorems by using Weyl function and spectral data which are called eigenvalues and normalizing constants for solution of inverse problem.

Description: The existence of solutions forp-Laplacian boundary value problem at resonance on the half-line is investigated. Ouranalysis relies on constructing the suitable Banach space,defining appropriate operators and using the extension of Mawhin'scontinuation theorem. An example is given to illustrate our mainresult.

Description: The authors of this paper study a nonlinear viscoelastic equation with variable exponents. By using the Faedo-Galerkin's method and embedding theory, the existence of weak solutions is given to the initial and boundary value problem under suitable assumptions.

Description: In this paper the inverse problem of finding the time-dependent coefficient of heat capacity together with the nonlocal boundary conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example.

Description: We study the following elliptic equations with variable exponents\begin{equation*}-\text{div}(\phi(x,|\nabla u|)\nabla u)=\lambda f(x,u)\quad \textmd{in }\quad \Bbb R^{N}.\end{equation*}Under suitable conditions on $\phi$ and $f$, we show theexistence of positivity of the infimum of all eigenvalues for theabove problem and then give some example to demonstrate our main result.

Description: Approximate symmetries of a mathematical model describingone-dimensional motion in a medium with a small nonlinearviscosity are studied. In a physical application, the approximatesolution is calculated making use of the approximate generator ofthe first-order approximate symmetry.

Description: In this note we consider the classical Massera Theorem, which proves the uniqueness of a periodic solution for the Lienard equationx''+f(x)x'+x=0and investigate the problem of existence of such periodic solution when f is monotone increasing for x〉0 and monotone decreasing for x

Description: No description available

Description: In this paper it is proved that both oscillatory integral operators and fractional oscillatory integral operators are bounded on generalized Morrey spaces $M_{p,\varphi}$. The correspondingcommutators generated by BMO functions are also considered.

Description: We develop a formulation for the analytic or approximate solution of Fractional differential equations (FDEs) by using respectively theanalytic or approximate solution of the differential equation,obtained by making fractional order of the original problem integer order. It is shown that this method works for FDEs very well. The results reveal that it is very effective and simple in determination of solutions of FDEs.

Description: The aim of this work is to study adaptive fully-discrete finiteelement methods forquadratic boundary optimal control problems governed bynonlinear parabolic equations. Wederive a posteriori error estimates for the state andcontrol approximation. Such estimates can be used to construct reliableadaptive finite element approximation for nonlinear quadratic parabolic boundary optimal controlproblems. Finally, we present anumerical example to show the theoretical results.

Description: \noindent{\bf Abstract:} This work is concerned with theexistence and uniqueness of positive solutions for the followingfractional boundary value problem:$$\left\{\begin{array}{l} -D^{\nu}_{0^+}y(t)=f(t,y(t),y(t))+g(t,y(t)), 0

Description: This paper investigates the existence of solutions forfractional differential inclusions of order $q \in (2,3]$with anti-periodic type integral boundary conditionsby means of some standard fixed point theorems for inclusions. Our results include the cases whenthe multivalued map involved in the problem has convex as well non-convex values. The paper concludes with an illustrative example.

Description: In this paper, given $f:\left[ a,b\right] \times \left( C\left( \left[ a,b%\right] \right) \right) ^{n-2}\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ a$L^{1}-$Carath\'{e}odory function, it is considered the functional higherorder equation\begin{equation*}u^{(n)}\left( x\right) =f(x,u,u^{\prime },...,u^{\left( n-2\right) }\left(x\right) ,u^{\left( n-1\right) }\left( x\right) )\end{equation*}%together with the nonlinear functional boundary conditions, for \ $%i=0,...,n-2$%\begin{equation*}\begin{array}{l}L_{i}(u,u^{\prime },...,u^{\left( n-1\right) },u^{\left( i\right) }\left(a\right) )=0,\text{ } \\L_{n-1}(u,u^{\prime },...,u^{\left( n-1\right) },u^{\left( n-2\right)}\left( b\right) )=0,%\end{array}%\end{equation*}Here $L_{i}$, $i=0,...,n-1$, are continuous functions. It will be proved anexistence and location result in presence of not necessarily ordered lowerand upper solutions, without assuming any monotone properties on theboundary conditions and on the nonlinearity $f.$

Description: In this paper we study thefollowing second-order periodic system:x''+V'(x)+p(t)|x|^{\alpha}=0where V(x) has a singularity.Under some assumptions on the V(x) and p(t)byOrtega' small twist theorem, we obtain theexistence of quasi-periodic solutions andboundedness of all the solutions .

Description: Recently, attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical). In a recent paper we studied a class of lattice sums that amount to solutions of Poisson's equation. By virtue of striking connections with Jacobi $\vartheta$-function values, we were able to develop new closed forms for certain solutions, and to extend such analysis to related lattice sums.We also alluded to results for the \emph{compressed sum}\begin{eqnarray}\label{phid:def1}\phi_2(x,y,d) &:=& \frac1{\pi^2} \sum_{m,n \in {\mathbb O}^2} \frac{\cos(\pi m x) \cos(\pi n\sqrt d \, y)}{m^2 + d n^2},\end{eqnarray}where $d〉0, x,y$ are real numbers. In this paper we first survey the earlier work and then discuss the sum \eqref{phid:def1} more completely..

Description: In this paper, the existence and blow-up property ofsolutions to initial and boundary value problem for a nonlinear parabolic system with variable exponents is studied. Meanwhile, the blow-up property of solutions for a nonlinear hyperbolic system is also obtained.

Description: The effects of a homogeneous-heterogeneous reaction on steady micropolar fluid flow from a permeable stretching or shrinking sheet in a porous medium are numerically investigated in this paper. The model developed by Chaudhary and Merkin (1995) for homogeneous-heterogeneous reaction in boundary layer flow with equaldiffusivities for reactant and autocatalyst is used and extended in this study. The uniqueness of this problem lies in the fact that the solutions are possible for all values of the stretching parameter lambda〉 0 while for lambda 〈 0 (shrinking surface), solutions are possible only for a limited range of values. The effects of physical and fluid parameters such as the stretching parameter, micropolar parameter, permeability parameter, Schmidt number, strength of homogeneous and heterogeneous reaction parameter on the skin friction, velocity and concentration are analyzed and these results are presented through graphs. The solute concentration at the surface is found to decrease with the strength of the homogeneous reaction, and to increase with heterogeneous reactions, the permeability parameter and stretching or shrinking parameters. The velocity at the surface was found increase with the micropolar parameter.

Description: In this paper, the author discusses the multiple positive solutions for an infinite three-point boundary value problem of first order impulsive superlinear singular integro-differential equations on the half linein a Banach space by means of the fixed point theorem of cone expansion and compression with norm type.

Description: In the unbounded domain we consider a problem with conditions given on inner characteristics in hyperbolic part of considered domain and on some parts of the line of parabolic degeneracy. We prove the unique solvability of the mentioned problem with help of the extremum principle. The proof of the solvability is based on the theory of singular integral equations, Wiener-Hopf equations, and Fredholm integral equations.

Description: In this paper, the separability properties of elliptic convolution operator equations are investigated. It is obtained that the corresponding convolution-elliptic operator is positive and also is a generatorof an analytic semigroup. By using these results the existence and uniqueness of maximal regular solution of the nonlinear convolution equation is obtained in Lp spaces. In application, maximal regularityproperties of anisotropic elliptic convolution equations are studied.

Description: The wave equation with boundary source term and fractional boundary dissipation is considered. The exponential growth for sufficiently large initial data is proved. To this end some techniques based on Fourier transforms and some inequalities such as the Hardy-Littlewood-Soblev inequality are used.

Description: The main purpose of this paper is to establish the existence, uniqueness and positive solution of a system of second-order boundary value problem with integral conditions. Using Banach’s fixed point theorem and the Leray-Schauder nonlinear alternative, we discuss the existence and uniqueness solution of this problem, and we apply Guo-Krasnoselskii’s fixed point theorem in cone to study the existence of positive solution. We also give some examples to illustrate our results.

Description:    We show that the Foldy-Wouthuysen transformation and its generalizations are simplified if the methods of pseudodifferential operators are used, which also allow estimating the exactness of the transition from the Dirac equation to the reduced equations for electrons and positrons. The methods and techniques used can be useful not only in studying asymptotic solutions of the Dirac equation but also in other problems. Content Type Journal Article Pages 547-566 DOI 10.1007/s11232-011-0041-y Authors J. Brüning, Humboldt University, Berlin, Germany V. V. Grushin, Moscow State Institute for Electronics and Mathematics, Moscow, Russia S. Yu. Dobrokhotov, Moscow Institute for Physics and Technology, Moscow, Russia T. Ya. Tudorovskii, Institute for Molecules and Materials Radboud, University of Nijmegen, Nijmegen, The Netherlands Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We present the bi-Hamiltonian structure and Lax pair of the equation ρ t = bu x +( 1/2 )[(u 2 −u x 2 )ρ] x , where ρ = u − u xx and b = const , which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions. Content Type Journal Article Pages 584-589 DOI 10.1007/s11232-011-0044-8 Authors Zhijun Qiao, Department of Mathematics, The University of Texas-Pan American, Edinburg, Texas, USA Xianqi Li, Department of Mathematics, The University of Texas-Pan American, Edinburg, Texas, USA Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We study the Wigner function in noncommutative quantum mechanics. By solving the time-independent Schrödinger equation on both a noncommutative space and a noncommutative phase space, we obtain the Wigner function for the Landau problem on those spaces. Content Type Journal Article Pages 628-635 DOI 10.1007/s11232-011-0047-5 Authors S. Dulat, School of Physics Science and Technology, Xinjiang University, Urumqi, China Kang Li, Department of Physics, Hangzhou Normal University, Hangzhou, China Jianhua Wang, Department of Physics, Shaanxi University of Technology, Hanzhong, China Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We review several purely mathematical results concerning boundary value problems for nonlinear pseudodifferential equations for p-adic closed and open strings in the tree approximation in the case d = 1 . For the solutions of these problems, we present formulas establishing the relations between the numbers of their zeros, the multiplicities of the zeros, and the numbers indicating how many times the solutions change sign. Content Type Journal Article Pages 539-546 DOI 10.1007/s11232-011-0040-z Authors V. S. Vladimirov, Steklov Mathematical Institute, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    Based on number theory, we present a new concept of gas without the particle interaction taken into account in which there are first-order phase transitions for T 〈 T cr on isotherms. We present formulas for new ideal gases, solving the Gibbs paradox, and also formulas for the transition to real gases based on the concept of the Zeno line. Content Type Journal Article Pages 654-667 DOI 10.1007/s11232-011-0050-x Authors V. P. Maslov, Lomonosov Moscow State University, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We establish that by choosing a smooth local perturbation of the boundary of a planar quantum waveguide, we can create an eigenvalue near any given threshold of the continuous spectrum and the corresponding trapped wave exponentially decaying at infinity. Based on an analysis of an auxiliary object, a unitary augmented scattering matrix, we asymptotically interpret Wood’s anomalies, the phenomenon of fast variations in the diffraction pattern due to variations in the near-threshold wave frequency. Content Type Journal Article Pages 606-627 DOI 10.1007/s11232-011-0046-6 Authors S. A. Nazarov, Institute of Mechanical Engineering Problems, RAS, St. Petersburg, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    Starting from the matrix KP hierarchy and adding a new τ B flow, we obtain a new extended matrix KP hierarchy and its Lax representation with the symmetry constraint on squared eigenfunctions taken into account. The new hierarchy contains two sets of times t A and τ B and also eigenfunctions and adjoint eigenfunctions as components. We propose a generalized dressing method for solving the extended matrix KP hierarchy and present some solutions. We study the soliton solutions of two types of ( 2+1 )-dimensional AKNS equations with self-consistent sources and two types of Davey-Stewartson equations with selfconsistent sources. Content Type Journal Article Pages 590-605 DOI 10.1007/s11232-011-0045-7 Authors Yehui Huang, Department of Mathematical Science, Tsinghua University, Beijing, China Xiaojun Liu, Department of Applied Mathematics, China Agricultural University, Beijing, China Yuqin Yao, Department of Applied Mathematics, China Agricultural University, Beijing, China Yunbo Zeng, Department of Mathematical Science, Tsinghua University, Beijing, China Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We propose a model on the Cayley tree and prove that a uncountable set of Ĝ-periodic Gibbs measures exists for this model, in contrast to models studied previously. Content Type Journal Article Pages 668-679 DOI 10.1007/s11232-011-0051-9 Authors U. A. Rozikov, Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences, Tashkent, Uzbekistan G. T. Madgoziev, Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences, Tashkent, Uzbekistan Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We propose a construction of a Lagrangian torus fibration of the full flag variety in ℂ 3 . In contrast to the classical fibration obtained from the Gelfand-Zeitlin system, the proposed fibration is special Lagrangian. Content Type Journal Article Pages 567-576 DOI 10.1007/s11232-011-0042-x Authors N. A. Tyurin, Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We propose a method for describing quasiequilibrium processes in a quasistable anharmonic oscillator affected by heat noise. We introduce effective thermodynamic parameters analogous to those of an equilibrium system but differing from them in numerical value. We define the parameter characterizing the nonequilibrium of the system, prove the mathematical consistency of the proposed method, and obtain the quasiequilibrium analogue of the theorem on energy distribution over degrees of freedom. Content Type Journal Article Pages 636-644 DOI 10.1007/s11232-011-0048-4 Authors A. V. Kargovsky, Lomonosov Moscow State University, Moscow, Russia L. S. Bulushova, Lomonosov Moscow State University, Moscow, Russia O. A. Chichigina, Lomonosov Moscow State University, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We develop the recent proposal to use dimensional reduction from the four-dimensional space-time (D = 1 + 3 ) to the variant with a smaller number of space dimensions D = 1 + d, d 〈 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation with a few toy examples (“educational toys”) of a space-time with a variable spatial geometry including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique for transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-type equation with potential generated by this variation is useful. We draw the following conclusions: ( 1 ) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). ( 2 ) The specific spectrum of scalar excitations resembles the spectrum of real particles; it reflects the geometry of the transition region and represents its “fingerprints.” ( 3 ) The parity violation due to the asymmetric character of the construction of our models could be related to the CP symmetry violation. Content Type Journal Article Pages 680-691 DOI 10.1007/s11232-011-0052-8 Authors P. P. Fiziev, Sofia University St. Kliment Ohridski, Sofia, Bulgaria D. V. Shirkov, Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    In the mode-coupling approximation, we consider the transition to the glass state in a system of collapsing hard spheres (a system with the hard-core potential to which a repulsive step is added). We propose an approximation for the structure factor of the system, which we use to construct the phase diagram of the transition to the glass state. We show that there exists a maximum on the liquid-glass curve corresponding to the reentrant transition to the glass state in the system. In the framework of the proposed model, we consider bifurcations of solutions of the equations describing the transition to the glass state and show that there exist bifurcations of the “swallow-tail” type corresponding to the glass-glass transition. Content Type Journal Article Pages 645-653 DOI 10.1007/s11232-011-0049-3 Authors V. N. Ryzhov, Institute for High Pressure Physics, RAS, Troitsk, Moscow Oblast, Russia E. E. Tareyeva, Institute for High Pressure Physics, RAS, Troitsk, Moscow Oblast, Russia Yu. D. Fomin, Institute for High Pressure Physics, RAS, Troitsk, Moscow Oblast, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We consider two initial boundary value problems on an interval with homogeneous Dirichlet conditions. These problems were proposed by M. I. Rabinovich and D. I. Trubetskov and are given by nonlinear fourth-order Sobolev-type equations. We prove the local-in-time existence of a strong generalized solution of one problem, and for both problems, we obtain sufficient conditions for the destruction of their strong generalized solutions in a finite time. Content Type Journal Article Pages 577-583 DOI 10.1007/s11232-011-0043-9 Authors M. O. Korpusov, Lomonosov Moscow State University, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We consider the evolution of the spatial distribution of fast atomic particles as they pass through a crystal. At small penetration depths where the particles move without significant loss of coherence, the joint probability density function becomes smoother at the expense of the gradient of the anharmonic potential of the planar channel. We show that as a result of this smoothing, there arises a quasiequilibrium (quasistationary) state of the subsystem of fast particles. The further evolution of the particle distribution is caused by nonelastic scattering and is described by the kinetic equation. At this stage, the anharmonic character of particle oscillations between the channel walls results in a renormalization of the total phonon scattering cross section of particles, and the renormalization is determined by the fourth-order anharmonic interaction. Content Type Journal Article Pages 506-516 DOI 10.1007/s11232-011-0038-6 Authors Yu. A. Kashlev, Baikov Institute for Metallurgy and Material Science, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We propose a method for calculating vacuum means of the scalar field energy-momentum tensor on Lie groups and homogeneous spaces. We use the generalized harmonic analysis based on the method of coadjoint representation orbits. Content Type Journal Article Pages 468-483 DOI 10.1007/s11232-011-0035-9 Authors A. I. Breev, Tomsk State University, Tomsk, Russia I. V. Shirokov, Omsk State Technical University, Omsk, Russia A. A. Magazev, Irtysh Branch of Novosibirsk State Academy of Water Transport, Novosibirsk, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We construct the group bundle for the inverse Fibonacci semigroups in the axiomatic approach framework. The proper Fibonacci semigroup is the corresponding group bundle for the Penrose semigroups, which can be interpreted as the generating grammar of the morphogenetic synthesis of the pentasymmetric Penrose parquet in the numbers of tiling by golden rhombuses. This morphogenetic synthesis of the Penrose parquet satisfies the scaling principle. Parquet plates are not absolutely rigid, and the relations between their metric characteristics are governed by the golden section and other magic numbers. The characteristic form factors of three-level dual alphabets are the corresponding invariants. We realize the morphogenetic synthesis in the examples of square-octagonal and bihexagonal lattices. We consider cumulative properties of magic series and the evolutionary aspects of semigroup orbits in the entropy representation. Content Type Journal Article Pages 517-537 DOI 10.1007/s11232-011-0039-5 Authors V. V. Yudin, Far-East State University, Vladivostok, Russia E. S. Startzev, Far-East State University, Vladivostok, Russia I. G. Permyakova, Far-East State University, Vladivostok, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We obtain an expression for the NMR line at low temperatures for a system of nuclear spins described by a Hamiltonian with equal spin-spin coupling constants. We show that in the case of “easy axis” anisotropy, the line has a logarithmic low-frequency singularity and an exponentially decreasing high-frequency asymptotic behavior at the temperature of an anomalous peak of heat capacity. In the case of “easy plane” anisotropy, the line has the traditional Gaussian form. We discuss the possibility of using NMR data to discover specific thermodynamic and magnetic properties of the considered model system. Content Type Journal Article Pages 496-505 DOI 10.1007/s11232-011-0037-7 Authors A. A. Khamzin, Kazan State Energy University, Kazan, Russia R. R. Nigmatullin, Kazan (Volga Region) Federal University, Kazan, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations. Content Type Journal Article Pages 421-443 DOI 10.1007/s11232-011-0033-y Authors A. V. Mikhailov, Applied Mathematics Department, University of Leeds, Leeds, UK Jing Ping Wang, School of Mathematics and Statistics, University of Kent, Kent, UK P. Xenitidis, Applied Mathematics Department, University of Leeds, Leeds, UK Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description: This paper gives growth properties of Green-Sch potentials at infinity in a cone, which generalizes results obtained by Qiao-Deng. The proof is based on the fact that the estimations of Green-Sch potentials with measures are connected with a kind of densities of the measures modified by the measures.MSC:35J10, 35J25.

Description: By using a variational method and some critical points theorems, we establish some results on the multiplicity of solutions for second-order impulsive differential equation depending on two real parameters on the half-line. In addition, two examples to illustrate our results are given.

Description: In this paper, we are concerned with a nonlinear coupled viscoelastic wave equations with initial-boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on relaxation functions, damping terms, and source terms, by using the energy method we proved a global nonexistence result for certain solutions with negative initial energy.

Description:    We propose a Lie-Noether-symmetry solution of two problems that arise with classical quantization: the quantization of higher-order (more than second) Euler-Lagrange ordinary differential equations of classical mechanics and the quantization of any second-order Euler-Lagrange ordinary differential equation that classically comes from a simple linear equation via nonlinear canonical transformations. Content Type Journal Article Pages 994-1001 DOI 10.1007/s11232-011-0081-3 Authors M. C. Nucci, Dipartimento di Matematica e Informatica, Università di Perugia and INFN, Sezione di Perugia, Perugia, Italy Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 168 Journal Issue Volume 168, Number 1

Description: Nonlinear physics: Theory and experiment VI Content Type Journal Article Pages 857-857 DOI 10.1007/s11232-011-0068-0 Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 168 Journal Issue Volume 168, Number 1

Description:    We consider the propagation of electromagnetic pulses in isotropic media taking a third-order nonlinearity into account. We develop a method for transforming Maxwell’s equations based on a complete set of projection operators corresponding to wave-dispersion branches (in a waveguide or in matter) with the propagation direction taken into account. The most important result of applying the method is a system of equations describing the one-dimensional dynamics of pulses propagating in opposite directions without accounting for dispersion. We derive the corresponding self-action equations. We thus introduce dispersion in the media and show how the operators change. We obtain generalized Schäfer-Wayne short-pulse equations accounting for both propagation directions. In the three-dimensional problem, we focus on optic fibers with dispersive matter, deriving and numerically solving equations of the waveguide-mode interaction. We discuss the effects of the interaction of unidirectional pulses. For the coupled nonlinear Schrödinger equations, we discuss a concept of numerical integrability and apply the developed calculation schemes. Content Type Journal Article Pages 974-984 DOI 10.1007/s11232-011-0079-x Authors M. Kuszner, Gdansk University of Technology, Gdansk, Poland S. Leble, Gdansk University of Technology, Gdansk, Poland B. Reichel, Gdansk University of Technology, Gdansk, Poland Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 168 Journal Issue Volume 168, Number 1

Description:    We study properties of the purely solitonic τ-function and potential of the heat equation in detail. We describe the asymptotic behavior of the potential and establish the ray structure of this asymptotic behavior on the plane (x 1 , x 2 ) in dependence on the parameters of the potential. Content Type Journal Article Pages 865-874 DOI 10.1007/s11232-011-0070-6 Authors M. Boiti, Dipartimento di Fisica, Università del Salento and INFN, Sezione di Lecce, Lecce, Italy F. Pempinelli, Dipartimento di Fisica, Università del Salento and INFN, Sezione di Lecce, Lecce, Italy A. K. Pogrebkov, Steklov Mathematical Institute, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 168 Journal Issue Volume 168, Number 1

Description: It is well-known that the second eigenvalue _2 of the Dirichlet Laplacian on the ball is not radial. Recently, Bartsch, Weth and Willem proved that the same conclusion holds true for the so-called nontrivial (sign changing) Fucik eigenvalues on the first curve of the Fucik spectrum which are close to the point (_2; _2). We show that the same conclusion is true in dimensions 2 and 3 without the last restriction.

Description: This paper is concerned with the existence of solutions to a class of p-Kirchho(R) type equationswith Neumann boundary data as follows:([M(Rjrujpdx)]p14pu = f(x; u); in ;@u@o = 0; on @.By means of a direct variational approach, we establish conditions ensuring the existence andmultiplicity of solutions for the problem.

Description: The electrogravitational instability of a dielectric oscillating streaming fluid cylinder surrounded by tenuous medium of negligible motion pervaded by transverse varying electric field has been investigated for all perturbation modes. The model is governed by Mathieu second order integro-differential equation. Some limiting cases are recovering from the present general one. The self-gravitating force is destabilizing only in the axisymmetric perturbation for long wavelengths, while, the axial electric field interior the fluid has strong destabilizing effect for all short and long wavelengths. The transverse field is strongly stabilizing. In the case of non-axisymmetric perturbation, the self-gravitating force is stabilizing for short and long waves, while, the electric field has stabilizing effect on short waves.

Description: For complete abstract with equations please see published manuscript.

Description: This paper investigates the blow-up properties of positive solutions to a system of evolution p-Laplace equations with nonlocal sources and inner absorptions. Under appropriate hypotheses, the authors discuss the global existence and blow-up of positive weak solutions by using a comparison principle.

Description: In this paper, we consider the initial-boundary value problem of one-dimensional compressible magnetohydrodynamics flows. The existence and continuous dependence of global solutions in $H^1$ have been established in Chen and Wang (Z Angew Math Phys 54, 608{632, 2003). We will obtain the regularity of global solutions under certain assumptions imposed on the initial data by driving some new a priori estimates.

Description:    We briefly review the half-a-century period (1959–2009) of the creation, development, and application of the method of two-time temperature retarded and advanced Green’s functions, introduced by Bogoliubov and Tyablikov in 1959. This method has not exhausted its resources by far, as is evidenced by its effective modification proposed by Plakida in 1970 and Tserkovnikov in 1971 and then, in a much more advanced form, by Tserkovnikov in 1981. Currently, a self-consistent description of both one-particle (transverse) and collective (longitudinal) Green’s functions and correlation functions is achieved based on the Green’s function method in the Bogoliubov-Tyablikov-Plakida-Tserkovnikov formulation. Content Type Journal Article Pages 1318-1329 DOI 10.1007/s11232-011-0108-9 Authors Yu. G. Rudoy, Peoples’ Friendship University of Russia, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 168 Journal Issue Volume 168, Number 3

Description:    We apply the Bogoliubov-Tyablikov method of two-time Green’s functions to a multielectron theory of metals in the framework of the models with strong correlations, which are the Hubbard model and the Vonsovskii s-d(f) exchange model. We discuss the atomic representation in the problem of describing electron states, the metal-insulator transition, the problem of noncoherent states, and the Kondo effect. We analyze the ferromagnetism criterion in systems of strongly correlated electrons, which differs drastically from the Stoner condition. Content Type Journal Article Pages 1246-1257 DOI 10.1007/s11232-011-0102-2 Authors V. Yu. Irkhin, Institute of Metal Physics, Ural Branch, RAS, Ekaterinburg, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 168 Journal Issue Volume 168, Number 3

Description: We study the multiplicity of periodic solutions of a class of non-autonomous delay differential equations. By making full use of the Clark dual, the dual variational functional is considered. Some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions.

Description: In this work we consider thesystem: \noindent\begin{eqnarray*}\left\{\begin{array}{ll}-\triangle_{p}u = \lambda [g(x)a(u)+f(v)]\qquad\qquad\qquad\qquad\qquad\qquad in~ \Omega\quad\vspace{0.3cm}\\-\triangle_{q}v =\lambda [g(x)b(v)+h(u)]\qquad\qquad\qquad\qquad \qquad \qquad in~\Omega \vspace{0.3cm}\\\qquad u= v=0~~~~~~~~\qquad\qquad\qquad\qquad\qquad\qquad\qquad~on~~\partial\Omega~,\end{array}\right.\end{eqnarray*}\underline{ where $\Omega$ is a bounded regionin $R^N$with smooth boundary $\partial\Omega,$ and $\triangle_p$ isthe p-laplacian operator defined by}\\\underline{ $\triangle_p {u}=div (|\nabla u|^{p-2}\nabla u) $ , $p,q〉1$ and $g(x)$ is a $C^{1}$sign-changing the weight function,that maybe negative}\\\underline{near the boundary.$f,h, a, b $are $C^{1}$ non-decreasing functions satisfying $a(0)\geq0, b(0)\geq 0 .$ By using the method }\\\underline{ of sub-supersolutions we prove the existence of weak solution.}( see manuscript for full abstract).

Description: In this paper, we investigate the effect of the coefficient f(z) of the subcritical nonlinearity. For sufficiently large lambda〉0, there are at least k+1 positive solutions of the semilinear elliptic equations.(Please see manuscript for full abstract).

Description: In the last decades, nonlocal boundary value problems have become a rapidly growing area of research. The study of this type of problems is driven not only by a theoretical interest, but also by the fact that several phenomena in engineering, physics and life sciences can be modelled in this way. For example, problems with feedback controls such as the steady-states of a thermostat, where a controller at one of its ends adds or removes heat, depending upon the temperature registered in another point, can be interpreted with a second-order ordinary differential equation subject to a three-point boundary condition.

Description: In this article, we explore the use of a non-perturbation linearization method to solve the coupled highlynonlinear system of equations due to flow over a vertical surface subject to a magnetic field. The lineariza-tion method is used in combination with an asymptotic expansion technique. The effects of Dufour, Soretand magnetic field parameters are investigated. The velocity, temperature and concentration distributionsas well as the skin-friction, heat and mass transfer coefficients have been obtained and discussed for vari-ous physical parametric values. The accuracy of the solutions has been tested using a local non-similarityMethodThe results show that the non-perturbation technique is an accurate numerical algorithm thatconverges rapidly and may serve as a viable alternative to finite difference and finite element methodsfor solving nonlinear boundary value problems.Mathematics Subject Classication (2000): 76D05; 74S30; 76E06; 76M25.

Description:    We review the problem of ambiguity in quantizing non-Abelian gauge theories and propose a method for quantizing them unambiguously. Content Type Journal Article Pages 198-202 DOI 10.1007/s11232-012-0021-x Authors A. A. Slavnov, Steklov Mathematical Institute, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 170 Journal Issue Volume 170, Number 2

Description:    Motivated by claims of broken rotational invariance in the WMAP data, a number of models have appeared in the literature realizing this effect through vector field(s) with a nonvanishing spatial vacuum expectation value. We discuss why many such models have ghost instabilities. Content Type Journal Article Pages 181-186 DOI 10.1007/s11232-012-0019-4 Authors M. Peloso, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota, USA Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 170 Journal Issue Volume 170, Number 2

Description: In memory of A. N. Tavkhelidze Content Type Journal Article Pages 131-131 DOI 10.1007/s11232-012-0014-9 Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 170 Journal Issue Volume 170, Number 2

Description:    We discuss a two-component liquid model for the quark-gluon plasma. We show that the model explains the basic experimental observations of the plasma properties naturally on the qualitative level. From the standpoint of the dynamics, the model assumes the existence of an effective scalar field with a nonzero vacuum expectation value. The hypothesis that such a condensate exists is supported by lattice data. We formulate the kind of lattice data that would yield a possible verification of the model. Content Type Journal Article Pages 211-216 DOI 10.1007/s11232-012-0023-8 Authors M. N. Chernodub, Laboratoire de Mathématiques et Physique Théorique, Université Francois-Rabelais de Tours, Tours, France H. Verschelde, Department of Physics and Astronomy, Ghent University, Ghent, Belgium V. I. Zakharov, Institute of Theoretical and Experimental Physics, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 170 Journal Issue Volume 170, Number 2

Description: In this article we propose approximation schemes for solving nonlinear initial boundary value problem with Volterra operator. Existence, uniqueness of solution as well as some regularity results are obtained via Rothe-Galerkin method.

Description: In this paper, the author investigates the existence and multiplicity of positive solutions for boundary value problem for fractional differential equation with $p$-Laplacian operator\[\left\{ {\begin{array}{l}D_{0+}^\beta (\varphi _p (D_{0+}^\alpha u))(t)+f(t,u(t))=0,0

Description: In this paper, we study the multiplicity of positive doubly periodic solutions for a singular semipositone telegraph equation. The proof is based on a well known fixed pointtheorem in a cone.

Description: In this paper, we discuss the existence of positive solutions to the singular Dirichlet boundary value problems (BVPs) in ordinary differential equations (ODEs) of the form\[u''(t) + \frac{a}{t}u'(t) - \frac{a}{t^2}u(t) = f(t,u(t),u'(t)), \quad u(0)=0, \enspace u(T)=0, \]where $a \in (-1,0)$. The nonlinearity $f(t,x,y)$ may be singular for the space variables $x=0$ and/or $y=0$. Moreover, since $a\not= 0$ the differential operator on the left hand side of the differential equation is singular at $t=0$. Sufficient conditions for the existence of positive solutions of the above BVPs are formulated and asymptotic properties of solutions are specified. The theory is illustrated by numerical experiments computed using the open domain {\sc Matlab} code {\tt bvpsuite}, based on polynomial collocation.

Description: In this study, the numerical solution of stagnation point flow over a stretching surface, with convective boundary conditions is considered. The transformed boundary layer equations are solved numerically using the Shooting method. Numerical solutions are obtained for the local heat transfer coefficient, the surface temperature as well as the velocity and temperature profiles. The features of the flow and heat transfer characteristics for various values of the Prandtl number, stretching parameter and conjugate parameter are analyzed and discussed.

Description: We study the existence of positive solutions of a nonlinear fractional heat equation with nonlocal boundary conditions depending on a positive parameter. Our results extend the second-order thermostat model to the non-integer case. We base our analysis on the known Guo-Krasnosel'skii fixed point theorem on cones.

Description: In this paper, we consider the existence of positive solutions for second order differential equations with deviating arguments and nonlocal boundary conditions. By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions under which such boundaryvalue problems have at least three positive solutions. We discuss our problem both for delayed and advanced arguments $\alpha$, and also in the case when $\alpha(t)=t, \; t\in [0,1]$. In all cases, the argument $\beta$ can change the character on $[0,1],$ see problem (1). It means that $\beta$ can be delayed in some set $\bar{J}\subset [0,1]$ and advanced in $[0,1]\setminus {\bar{J}}.$An example is added to illustrate the results.

Description: Based on the fixed point index and partial order method, one new order-type existence theorem concerning cone expansion and compression is established.As applications, we present sufficient existence conditions for the first and second order periodic problems.

Description: In this article, we study the periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions. By using the theory of Leray-Schauder degree, we obtain the existence of nontrivial nonnegative time periodic solution.

Description: In this paper, we investigate the complicated asymptotic behavior of the solutions to the Cauchy problem of the porous medium equation with nonlinear sources when the initial value belongs to a weighted $L^{\infty}$ space.

Description: Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper we use the derivative sampling theorem \lq\lq Hermite interpolations \rq\rq to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Using computable error bounds we obtaineigenvalue enclosures. Examples with tables and illustrative figures are given. Also Numerical examples, which are given at the end of the paper, give comparisons with the classical sinc-method in Annaby and Tharwat (BIT {\bf 47}, 699--713 (2007)) and explain that the Hermite interpolations method gives remarkably better results.

Description: In this paper, we study three point boundary value problemsof the following fractional functional differential equationsinvolving the Caputo fractional derivative:$$\begin{array}{ll}^CD^{\alpha}u(t)=f(t,u_{t},^CD^{\beta}{u(t)}),\; 0

Description:    We consider two versions of the scenario for generating scalar cosmological perturbations based on the conformal symmetry: a spectator version with a scalar field conformally coupled to gravity and with a negligible energy density; a dynamical version with a scalar field minimally coupled to gravity and dominating the cosmological evolution. Using the Newtonian gauge, we show, first, that no UV strong-coupling scale is generated below M Pl , because of mixing with metric perturbations in the dynamical scenario, and, second, that both the dynamical and the spectator models yield identical results in the leading nonlinear order. These results, which include potentially observable effects like statistical anisotropy and non-Gaussianity, hold for the entire class of conformal models. As an example, in the dynamical scenario with the comoving gauge, we reproduce our result on the statistical anisotropy, previously obtained in the framework of the spectator approach. Content Type Journal Article Pages 1457-1465 DOI 10.1007/s11232-012-0126-2 Authors M. V. Libanov, Institute for Nuclear Research, RAS, Moscow, Russia V. A. Rubakov, Institute for Nuclear Research, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We study the hard-core model on the Cayley tree. We show that this model admits only periodic Gibbs measures with the period two. We find sufficient conditions for all weakly periodic Gibbs measures to be translation invariant. Content Type Journal Article Pages 1377-1386 DOI 10.1007/s11232-012-0120-8 Authors U. A. Rozikov, Institute of Mathematics at National University of Uzbekistan, Tashkent, Uzbekistan R. M. Khakimov, Institute of Mathematics at National University of Uzbekistan, Tashkent, Uzbekistan Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter α, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of α. We consider Bose—Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 〈 α 〈 2 is greater in the case under consideration than in the case of an ideal system, where α = 2. Content Type Journal Article Pages 1445-1456 DOI 10.1007/s11232-012-0125-3 Authors Z. Z. Alisultanov, Prokhorov General Physics Institute, RAS, Moscow, Russia R. P. Meilanov, Institute of Geothermal Problems, Dagestan Scientific Center, RAS, Makhachkala, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We show that the Weyl correspondence and the concept of a Moyal multiplier can be naturally extended to generalized function classes that are larger than the class of tempered distributions. This generalization is motivated by possible applications to noncommutative quantum field theory. We prove that under reasonable restrictions on the test function space E ⊂ L 2 , any operator in L 2 with a domain E and continuous in the topologies of E and L 2 has a Weyl symbol, which is defined as a generalized function on the Wigner-Moyal transform of the projective tensor square of E. We also give an exact characterization of the Weyl transforms of the Moyal multipliers for the Gel’fand-Shilov spaces S β β . Content Type Journal Article Pages 1359-1376 DOI 10.1007/s11232-012-0119-1 Authors M. A. Soloviev, Lebedev Physical Institute, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We propose a uniform method for estimating fractal characteristics of systems satisfying some type of scaling principle. This method is based on representing such systems as generating Bethe-Cayley tree graphs. These graphs appear from the formalism of the group bundle of Fibonacci-Penrose inverse semigroups. We consistently consider the standard schemes of Cantor and Koch in the new method. We prove the fractal property of the proper Fibonacci system, which has neither a negative nor a positive redundancy type. We illustrate the Fibonacci fractal by an original procedure and in the coordinate representation. The golden ratio and specific inversion property intrinsic to the Fibonacci system underlie the Fibonacci fractal. This property is reflected in the structure of the Fibonacci generator. Content Type Journal Article Pages 1387-1402 DOI 10.1007/s11232-012-0121-7 Authors V. V. Yudin E. S. Startzev, Institute for Physics and Information Technologies, Far-East Federal University, Vladivostok, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We consider classical aspects of the dynamics of scalar fields with nonpositive-definite potentials in the Friedmann cosmology. Such models appear as effective local models in the framework of nonlocal models related to string field theory. In an effective local approximation after a suitable field redefinition, the Higgs potential with a negative effective cosmological constant appears in these models. A special feature of considered models is an absence of the reheating phase, instead of which phase of regime change from expansion to contraction appears. We study the behavior of the model near the point of regime change. Content Type Journal Article Pages 1466-1480 DOI 10.1007/s11232-012-0127-1 Authors I. Ya. Arefeva, Steklov Mathematical Institute, RAS, Moscow, Russia N. V. Bulatov, Lomonosov Moscow State University, Moscow, Russia R. V. Gorbachev, Steklov Mathematical Institute, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We construct a family of two-gap solutions of the focusing nonlinear Schrödinger equation and derive a condition under which the solutions behave as the so-called freak waves located at the nodes of a two-dimensional lattice. We also study how the lattice parameters depend on the parameters of the spectral curve. Content Type Journal Article Pages 1403-1416 DOI 10.1007/s11232-012-0122-6 Authors A. O. Smirnov, St. Petersburg State University of Aerospace Instrumentation, St. Petersburg, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We consider perturbations of a second-order periodic operator on the line; the Schrödinger operator with a periodic potential is a specific case of such an operator. The perturbation is realized by a potential depending on two small parameters, one of which describes the length of the potential support, and the inverse value of other corresponds to the value of the potential. We obtain sufficient conditions for the perturbing potential to have eigenvalues in the gaps of the continuous spectrum. We also construct their asymptotic expansions and present sufficient conditions for the eigenvalues of the perturbing potential to be absent. Content Type Journal Article Pages 1438-1444 DOI 10.1007/s11232-012-0124-4 Authors R. R. Gadylshin, Akmulla Bashkir State Pedagogical University, Ufa, Russia I. Kh. Khusnullin, Akmulla Bashkir State Pedagogical University, Ufa, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    We briefly review problems arising in the study of the beta deformation, which turns out to be the most difficult element in a number of modern problems: the deviation of β from unity is connected with the “exit from the free-fermion point” in two-dimensional conformal theories, from the symmetric graviphoton field with ∈ 2 = −∈ 1 in instanton sums in four-dimensional supersymmetric Yang-Mills theories, with the transition from matrix models to beta ensembles, from HOMFLY polynomials to superpolynomials in the Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras, and so on. We mainly attend to issues related to the Alday-Gaiotto-Tachikawa correspondence and its possible generalizations. Content Type Journal Article Pages 1417-1437 DOI 10.1007/s11232-012-0123-5 Authors A. Yu. Morozov, Institute for Theoretical and Experimental Physics, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description:    In the braided context, we rederive a popular nonsemisimple fusion algebra from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p, 1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an entwined category (i.e., a category with monodromy but not with braiding). Content Type Journal Article Pages 1329-1358 DOI 10.1007/s11232-012-0118-2 Authors A. M. Semikhatov, Lebedev Physical Institute, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 173 Journal Issue Volume 173, Number 1

Description: By properly constructing a functional and by using the criticalpoint theory, we establish the existence of homoclinic solutions fora class of subquadratic second-order Hamiltonian systems. Our resultgeneralizes and improves some existing ones. An example is given toshow that our theorem applies, while the existing results are not applicable.

Description: We tackle the existence and uniqueness of the solution for a kind of integro-differential equations involving the generalized p-Laplacian operator with mixed boundary conditions. This is achieved by using some results on the ranges for maximal monotone operators and pseudo-monotone operators. The method used in this paper extends and complements some previous work.

Description: We will consider the nonhomogeneous $\phi$-Laplacian differential equation\begin{equation*}\begin{cases}\left ( \phi\left (u'(t) \right ) \right )'=-h(t)f(u(t)),\;\ t\in (0,T),\\u(0)=\sum\limits_{i=1}^{k}\alpha_iu(\eta_i),\;\phi\left (u'(T) \right )=\beta,\end{cases}\end{equation*}where $\phi:\mathbb{R} \to (-b,b)(0

Description: The aim of this paper is to establish multiplicity results of non-trivial and nonnegative solutions for mixed boundary value problems with the Sturm-Liouville equation. The approach isbased on variational methods.

Description: In this paper, we consider some nonlinear pseudo-parabolic Benjamin-Bona-Mahony-Burgers (BBMB) equations. These equations are of a class of nonlinear pseudo-parabolic or Sobolev type equations u_{t}-deltau_{t}-alphadeltau=f(x,u,nablau),alpha is a fixed positive constant, arising from the mathematical physics. The tanh method with the aid of symbolic computational system is employed to investigate exact solutions of BBMB type equations and the exact solutions are found. The results obtained can be viewed as a verification and improvement of the previously known data.

Description: In this study, we establish some conditions for existence and uniqueness of the solutions to semilinear fractional impulsive integro-differential evolution equations with non-local conditions by using Schauder's fixed point theorem and contraction mapping principle.