Dipole and Quadrupole Solitons in Optically Induced Two-Dimensional Photonic Lattices: Theory and Experiment

Dipole and quadrupole solitons in a two-dimensional photorefractive optical lattice are investigated both theoretically and experimentally. It is shown theoretically that out-of-phase dipole solitons and quadrupole solitons exist and are linearly stable in the intermediate-intensity regime. In-phase dipole and quadrupole solitons, however, are always linearly unstable, but their instabilities are rather weak in the low-intensity regime. Experimentally, both types of dipole solitons are observed, and the experimental results agree qualitatively with the theoretical predictions. In addition, we have observed the anisotropic effect of the photorefractive crystal in the dipole-soliton formation.     

Elektrodynamik mit Spin

Atomistic equations of the electromagnetic field for a particle with spin are derived from a Lagrangian. These equations are consistent with the equations of motion for such a particle. The resulting phenomenological equations are the well-known equations of Maxwell for the electromagnetic field in matter. The atomistic field equations for a particle with spin and magnetic moment give a dipole field. This result and the corresponding quantum mechanics for a particle with spin are applied to compute the hyperfine structure of the hydrogen atom by perturbation theory.     

Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations

We establish one-to-one transformations and self-maps between nonlinear diffusion equations in nonhomogeneous media, where the density function is given by a power. We use these transformations to deduce new interesting self-similar, radially symmetric solutions of the equations. In particular, Barenblatt, dipole and focusing Aronson-Graveleau type solutions are deduced, and some equations with singular potentials are studied. The new solutions are example of interesting or unexpected mathematical features of these equations, providing also natural candidates for the asymptotic behavior.     

A transient electromechanical process in a liquid crystal dipole

On the basis of the equations of the electromagnetic field of an orthotropic medium and the Lagrange equations for rotational motion, we obtain the equations that describe the electromechanical state of a liquid crystal dipole in an external electric field. The results of computer modeling of the transient process when an electric field is abruptly imposed are given. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 160–162.     

Interaction of point and dipole vortices in an incompressible liquid

We discuss the interaction between point vortices and point dipole vortices in two-dimensional ideal hydrodynamics and show that the equations of motion of the interacting point and point dipole vortices are exactly integrable. We find exact solutions for all possible parameter values characterizing the vortices and for arbitrary initial conditions and establish the regimes of vortex motion.     

楔型向错偶极子和裂纹的干涉效应

研究了晶体材料中一个楔型向错偶极子与裂纹的弹性干涉效应．运用复变函数方法获得了复势函数和应力场的封闭形式解答,导出了裂纹尖端应力强度因子和作用在向错偶极子中心点像力的解析表达式．获得了向错偶极子的位置、方向和偶臂长度对裂纹尖端应力强度因子的影响规律,并讨论了裂纹附近向错偶极子的平衡位置．结果表明向错偶极子靠近裂纹尖端时,对应力强度因子有明显的屏蔽或反屏蔽作用．     

Soliton Interactions for the Three‐Coupled Discrete Nonlinear Schrödinger Equations in the Alpha Helical Proteins

Three‐coupled discrete nonlinear Schrödinger equations, which describe the dynamics of the three hydrogen bonding spines in the alpha helical proteins with the interspine coupling at the discrete level, are investigated. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation of those equations. Propagation characteristics and interactions of the bound‐state solitons are discussed. Bound states of two and three bright solitons arise when all of them propagate in parallel. Elastic interaction between the bound‐state solitons and one bright soliton is given. Increase of the dipole‐dipole interaction energy can lead to the increase of the soliton velocity, that is, the one‐interaction period becomes shorter.     

Continual approach to multiparticle systems with long-range interaction. Hierarchy of macroscopic fields and physical consequences

Macroscopic physical phenomena in multiparticle systems with long-range interaction are shown to be determined by the hierarchy of macroscopic fields with sources from different tensor ranks. A method of deriving continual equations in the (r, t) space is suggested. Equations describing the dynamics of the dipole moment and dipole macroscopic fields are obtained. It is shown that a more thorough consideration of the interactions may lead to essential physical consequences. The energy balance equation is derived for an infinitesimal volume as an open system. The heat flow is shown to be a flow of the total energy of this volume rather than a flow of the thermal kinetic energy.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 3 15, July, 1996.     

Index Reduction and Discontinuity Handling Using Substitute Equations

Several techniques exist for index reduction and consistent initialization of higher index DAEs. Many such techniques change the original set of equations by differentiation, substitution, and/or introduction of new variables. This paper introduces substitute equations as a new language element. By means of a substitute equation, the value of a continuous variable or its time derivative is specified by an expression. This expression is evaluated each time that the variable or its time derivative, respectively, is referenced in the model. The advantage of substitute equations is that they enable index reduction and consistent initialization of higher index DAEs without changing the original equations; no existing variables are removed and no new variables are introduced. Substitute equations can also be used to enable the use of general purpose numerical solvers for equations where one or more of the unknowns are discontinuous, and they can be used to prevent functions to be called outside of their domain.     

Gravitational Effects on Light Rays and Binary Pulsar Energy Loss in a Scalar Theory of Gravity

We summarize a scalar bimetric theory of gravity with a preferred reference frame. The dynamics are governed by an extension of Newton's second law. We recover geodesic motion together with Newton's attraction field in the static case and find Schwarzschild's metric in the static spherical case. We build asymptotic schemes of post-Newtonian (PN) and post-Minkowskian (PM) approximations, each based on associating a conceptual family of systems with the given system. At the 1PN approximation, there is no preferred-frame effect for photons, and we hence obtain the standard predictions of GR for photons. At the 0PM approximation, an isolated system loses energy by quadrupole radiation without any monopole or dipole term. Inserting this loss into the Newtonian two-body problem gives the Peters–Mathews coefficients of the theory.     

Time-space integrable decompositions of nonlinear evolution equations

Separation of the time and space variables of evolution equations is analyzed, without using any structure associated with evolution equations. The resulting theory provides techniques for constructing time-space integrable decompositions of evolution equations, which transform an evolution equation into two compatible Liouville integrable ordinary differential equations in the time and space variables. The techniques are applied to the KdV, MKdV and diffusion equations, thereby yielding several new time-space integrable decompositions of these equations.     

Flux-splitting schemes for parabolic equations with mixed derivatives

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.     

An two phase abs method for solving over determined systems of linear inequalities

The method, called the Multi-Stage ABS algorithm, for solving the over-determined linear inequalities system and the system combined with the over-determined linear inequalities and the equations is presented. This method is characterized by translating inequalities system to an equations system with slack variables. The explicit solution with the slack variables of the equations system are given by the implicit LU algorithm, then the slack variables can be given by the ABS algorithm. Finally, the upper multiplications of the algorithm are given.     

Amplitude–shape approximation as an extension of separation of variables

Separation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd.     

Biomagnetic fluid flow over a stretching sheet with non linear temperature dependent magnetization

The flow of a heated ferrofluid over a linearly stretching sheet is studied in the pres- ence of an applied magnetic field due to a magnetic dipole. It is assumed that the applied magnetic field is sufficiently strong to saturate the ferrofluid and the variation of magnetization with temperature can be approximated by a non linear function of temperature difference. By introducing appropriate non dimensional variables the problem is described by a coupled and non linear system of ordinary differential equations with its boundary conditions which is solved numerically by applying an efficient numerical technique based on the common finite difference method. The obtained results are presented graphically for different values of the parameters entering into the problem under consideration and the dependence of the flow field from these parameters is discussed. A comparative study, with a similar problem which has already been solved and documented in literature, is also made wherever necessary, emphasizing the impor- tance of the non-linear variation of magnetization with temperature. Emphasis is also given in the obtained results for Prandtl number equal to 21 and critical exponent = 0.368 which are important and interesting in Biomagnetic Fluid Dynamics.     

Numerical methods for nonlinear first‐order partial differential equations with deviated variables

In the article classical solutions of initial problems for nonlinear differential equations with deviated variables are approximated by solutions of quasilinear systems of difference equations. Interpolating operators on the Haar pyramid are used. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on a comparison method. This new approach to solving nonlinear equations with deviated variables numerically is based on a method of linearization for initial problems. Numerical examples are given. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Biomagnetic fluid flow over a stretching sheet with non linear temperature dependent magnetization

The flow of a heated ferrofluid over a linearly stretching sheet is studied in the pres- ence of an applied magnetic field due to a magnetic dipole. It is assumed that the applied magnetic field is sufficiently strong to saturate the ferrofluid and the variation of magnetization with temperature can be approximated by a non linear function of temperature difference. By introducing appropriate non dimensional variables the problem is described by a coupled and non linear system of ordinary differential equations with its boundary conditions which is solved numerically by applying an efficient numerical technique based on the common finite difference method. The obtained results are presented graphically for different values of the parameters entering into the problem under consideration and the dependence of the flow field from these parameters is discussed. A comparative study, with a similar problem which has already been solved and documented in literature, is also made wherever necessary, emphasizing the impor- tance of the non-linear variation of magnetization with temperature. Emphasis is also given in the obtained results for Prandtl number equal to 21 and critical exponent = 0.368 which are important and interesting in Biomagnetic Fluid Dynamics.     

Diophantine equations over global function fields IV: S-unit equations in several variables with an application to norm form equations

We describe an efficient algorithm to calculate all solutions of unit equations in several variables over global function fields. Note that using the present tools it is not possible to solve completely unit equations in more than two variables over number fields. In the function field case such equations are completely solved here for the first time. As a typical application we determine all solutions of norm form equations.