Advances in Mathematical Physics https://www.hindawi.com The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. Two Kinds of Classifications Based on Improved Gravitational Search Algorithm and Particle Swarm Optimization Algorithm Mon, 11 Sep 2017 09:41:33 +0000 http://www.hindawi.com/journals/amp/2017/2131862/ Gravitational Search Algorithm (GSA) is a widely used metaheuristic algorithm. Although fewer parameters in GSA were adjusted, GSA has a slow convergence rate. In this paper, we change the constant acceleration coefficients to be the exponential function on the basis of combination of GSA and PSO (PSO-GSA) and propose an improved PSO-GSA algorithm (written as I-PSO-GSA) for solving two kinds of classifications: surface water quality and the moving direction of robots. I-PSO-GSA is employed to optimize weights and biases of backpropagation (BP) neural network. The experimental results show that, being compared with combination of PSO and GSA (PSO-GSA), single PSO, and single GSA for optimizing the parameters of BP neural network, I-PSO-GSA outperforms PSO-GSA, PSO, and GSA and has better classification accuracy for these two actual problems. Hongping Hu, Xiaxia Cui, and Yanping Bai Copyright © 2017 Hongping Hu et al. All rights reserved. Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation Mon, 11 Sep 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/1743789/ Based on the symbolic computation, a class of lump solutions to the (2+1)-dimensional Sawada-Kotera (2DSK) equation is obtained through making use of its Hirota bilinear form and one positive quadratic function. These solutions contain six parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are free. Then by adding an exponential function into the original positive quadratic function, the interaction solutions between lump solutions and one stripe soliton are derived. Furthermore, by extending this method to a general combination of positive quadratic function and hyperbolic function, the interaction solutions between lump solutions and a pair of resonance stripe solitons are provided. Some figures are given to demonstrate the dynamical properties of the lump solutions, interaction solutions between lump solutions, and stripe solitons by choosing some special parameters. Xian Li, Yao Wang, Meidan Chen, and Biao Li Copyright © 2017 Xian Li et al. All rights reserved. Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method Thu, 07 Sep 2017 06:43:53 +0000 http://www.hindawi.com/journals/amp/2017/2610804/ The second-kind Chebyshev wavelets collocation method is applied for solving a class of time-fractional diffusion-wave equation. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the second kind. Moreover, convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two dimensions are given. During the process of establishing the expression of the solution, all the initial and boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Based on the collocation technique, the second-kind Chebyshev wavelets are used to reduce the problem to the solution of a system of linear algebraic equations. Several examples are provided to confirm the reliability and effectiveness of the proposed method. Fengying Zhou and Xiaoyong Xu Copyright © 2017 Fengying Zhou and Xiaoyong Xu. All rights reserved. Geometrical/Physical Interpretation of the Conserved Quantities Corresponding to Noether Symmetries of Plane Symmetric Space-Times Wed, 30 Aug 2017 10:18:42 +0000 http://www.hindawi.com/journals/amp/2017/4384093/ The aim of this paper is to give the geometrical/physical interpretation of the conserved quantities corresponding to each Noether symmetry of the geodetic Lagrangian of plane symmetric space-times. For this purpose, we present a complete list of plane symmetric nonstatic space-times along with the generators of all Noether symmetries of the geodetic Lagrangian. Additionally, the structure constants of the associated Lie algebras, the Riemann curvature tensors, and the energy-momentum tensors are obtained for each case. It is worth mentioning that the list contains all classes of solutions that have been obtained earlier during the classification of plane symmetric space-times by isometries and homotheties. Bismah Jamil, Tooba Feroze, and Andrés Vargas Copyright © 2017 Bismah Jamil et al. All rights reserved. The Perturbed Riemann Problem for Special Keyfitz-Kranzer System with Three Piecewise Constant States Tue, 29 Aug 2017 07:07:03 +0000 http://www.hindawi.com/journals/amp/2017/3276182/ The Riemann problem for a special Keyfitz-Kranzer system is investigated and then seven different Riemann solutions are constructed. When the initial data are chosen as three piecewise constant states under suitable assumptions, the global solutions to the perturbed Riemann problem are constructed explicitly by studying all occurring wave interactions in detail. Furthermore, the stabilities of solutions are obtained under the specific small perturbations of Riemann initial data. Yuhao Jiang and Chun Shen Copyright © 2017 Yuhao Jiang and Chun Shen. All rights reserved. Hamilton-Poisson Realizations of the Integrable Deformations of the Rikitake System Sun, 27 Aug 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/4596951/ Integrable deformations of an integrable case of the Rikitake system are constructed by modifying its constants of motions. Hamilton-Poisson realizations of these integrable deformations are given. Considering two concrete deformation functions, a Hamilton-Poisson approach of the obtained system is presented. More precisely, the stability of the equilibrium points and the existence of the periodic orbits are proved. Furthermore, the image of the energy-Casimir mapping is determined and its connections with the dynamical elements of the considered system are pointed out. Cristian Lăzureanu Copyright © 2017 Cristian Lăzureanu. All rights reserved. Robust Output Synchronization of Arrays of Chaotic Sprott Circuits Wed, 23 Aug 2017 09:18:45 +0000 http://www.hindawi.com/journals/amp/2017/5201634/ This article presents a technique for synchronizing arrays of a class of chaotic systems known as Sprott circuits. This technique can be applied to different topologies and is robust to parametric uncertainties caused by tolerances in the electronic components. The design of coupling signals is based on the definition of a set of functionals which depend on the errors between the outputs of the nodes and the errors between the output of a reference system and the outputs of the nodes. When there are no parametric uncertainties, we establish a criterion to design the coupling signals using only one state variable of each system. When the parametric uncertainties are present, we add a robust observer and a low pass filter to estimate the perturbation terms, which are subsequently compensated through the coupling signals, resulting in a robust closed loop system. The performance of the synchronization technique is illustrated by real-time simulations. Ernesto V. Gonzalez Solis and David I. Rosas Almeida Copyright © 2017 Ernesto V. Gonzalez Solis and David I. Rosas Almeida. All rights reserved. On Concrete Spectral Properties of a Twisted Laplacian Associated with a Central Extension of the Real Heisenberg Group Sun, 20 Aug 2017 09:33:52 +0000 http://www.hindawi.com/journals/amp/2017/7575820/ We consider the special magnetic Laplacian given by . We show that is connected to the sub-Laplacian of a group of Heisenberg type given by realized as a central extension of the real Heisenberg group . We also discuss invariance properties of and give some of their explicit spectral properties. Aymane El Fardi, Allal Ghanmi, and Ahmed Intissar Copyright © 2017 Aymane El Fardi et al. All rights reserved. Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE Thu, 17 Aug 2017 09:21:23 +0000 http://www.hindawi.com/journals/amp/2017/3204959/ The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem. Chunlong Sun, Gongsheng Li, and Xianzheng Jia Copyright © 2017 Chunlong Sun et al. All rights reserved. An Iterative Method for Solving of Coupled Equations for Conductive-Radiative Heat Transfer in Dielectric Layers Wed, 16 Aug 2017 09:45:34 +0000 http://www.hindawi.com/journals/amp/2017/9139135/ The mathematical model for describing combined conductive-radiative heat transfer in a dielectric layer, which emits, absorbs, and scatters IR radiation both in its volume and on the boundary, has been considered. A nonlinear stationary boundary-value problem for coupled heat and radiation transfer equations for the layer, which exchanges by energy with external medium by convection and radiation, has been formulated. In the case of optically thick layer, when its thickness is much more of photon-free path, the problem becomes a singularly perturbed one. In the inverse case of optically thin layer, the problem is regularly perturbed, and it becomes a regular (unperturbed) one, when the layer’s thickness is of order of several photon-free paths. An iterative method for solving of the unperturbed problem has been developed and its convergence has been tested numerically. With the use of the method, the temperature field and radiation fluxes have been studied. The model and method can be used for development of noncontact methods for temperature testing in dielectrics and for nondestructive determination of its radiation properties on the base of the data obtained by remote measuring of IR radiation emitted by the layer. Vasyl Chekurin and Yurij Boychuk Copyright © 2017 Vasyl Chekurin and Yurij Boychuk. All rights reserved. The Magic of Universal Quantum Computing with Permutations Tue, 15 Aug 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/5287862/ The role of permutation gates for universal quantum computing is investigated. The “magic” of computation is clarified in the permutation gates, their eigenstates, the Wootters discrete Wigner function, and state-dependent contextuality (following many contributions on this subject). A first classification of a few types of resulting magic states in low dimensions is performed. Michel Planat and Rukhsan Ul Haq Copyright © 2017 Michel Planat and Rukhsan Ul Haq. All rights reserved. Solutions of Navier-Stokes Equation with Coriolis Force Mon, 14 Aug 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/7042686/ We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, . In both steady state and two-dimensional flow, the vorticity vector gets shifted by the amount of . Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow , the equation satisfied by is independent of . The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of . Sunggeun Lee, Shin-Kun Ryi, and Hankwon Lim Copyright © 2017 Sunggeun Lee et al. All rights reserved. Niemeier Lattices in the Free Fermionic Heterotic–String Formulation Thu, 10 Aug 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/3572469/ The spinor–vector duality was discovered in free fermionic constructions of the heterotic string in four dimensions. It played a key role in the construction of heterotic–string models with an anomaly-free extra symmetry that may remain unbroken down to low energy scales. A generic signature of the low scale string derived model is via diphoton excess that may be within reach of the LHC. A fascinating possibility is that the spinor–vector duality symmetry is rooted in the structure of the heterotic–string compactifications to two dimensions. The two-dimensional heterotic–string theories are in turn related to the so-called moonshine symmetries that underlie the two-dimensional compactifications. In this paper, we embark on exploration of this connection by the free fermionic formulation to classify the symmetries of the two-dimensional heterotic–string theories. We use two complementary approaches in our classification. The first utilises a construction which is akin to the one used in the spinor–vector duality. Underlying this method is the triality property of representations. In the second approach, we use the free fermionic tools to classify the twenty-four-dimensional Niemeier lattices. Panos Athanasopoulos and Alon E. Faraggi Copyright © 2017 Panos Athanasopoulos and Alon E. Faraggi. All rights reserved. Elliptic Function Solutions in Jackiw-Teitelboim Dilaton Gravity Mon, 31 Jul 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/2154784/ We present a new family of solutions for the Jackiw-Teitelboim model of two-dimensional gravity with a negative cosmological constant. Here, a metric of constant Ricci scalar curvature is constructed, and explicit linearly independent solutions of the corresponding dilaton field equations are determined. The metric is transformed to a black hole metric, and the dilaton solutions are expressed in terms of Jacobi elliptic functions. Using these solutions, we compute, for example, Killing vectors for the metric. Jennie D’Ambroise and Floyd L. Williams Copyright © 2017 Jennie D’Ambroise and Floyd L. Williams. All rights reserved. An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two-Dimensional Volterra-Fredholm Integrodifferential Equations Sun, 30 Jul 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/9385040/ We propose and apply coupling of the variational iteration method (VIM) and homotopy perturbation method (HPM) to solve nonlinear mixed Volterra-Fredholm integrodifferential equations (VFIDE). In this approach, we use a new formula called variational homotopy perturbation method (VHPM) and variational accelerated homotopy perturbation method (VAHPM). This approach is based on the form of He’s polynomials and on a new form of He’s polynomials. We discuss the convergence of the technique. Some numerical examples are introduced to verify the efficiency of this technique. F. A. Hendi and M. M. Al-Qarni Copyright © 2017 F. A. Hendi and M. M. Al-Qarni. All rights reserved. Singular Limit of the Rotational Compressible Magnetohydrodynamic Flows Tue, 25 Jul 2017 09:34:12 +0000 http://www.hindawi.com/journals/amp/2017/9493186/ We consider the compressible models of magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We derive a rigorous quasi-geostrophic equation governed by magnetic field from the stratified flows of the rotational compressible magnetohydrodynamic flows with the well-prepared initial data and the tool of proof is based on the relative entropy. Furthermore, the convergence rates are obtained. Young-Sam Kwon Copyright © 2017 Young-Sam Kwon. All rights reserved. Anomalous Diffusion with an Irreversible Linear Reaction and Sorption-Desorption Process Tue, 25 Jul 2017 08:32:28 +0000 http://www.hindawi.com/journals/amp/2017/6361598/ We investigate the diffusion of two different species in a semi-infinite medium considering the presence of linear reaction terms. The dynamics for these species is governed by fractional diffusion equations. We also consider the presence of an adsorption-desorption boundary condition. The solutions for this system are found in terms of the function of Fox and by analyzing the behavior of the mean square displacement a rich class of diffusion processes is verified. In this sense, we show how the surface effects modify the bulk dynamics and promote an anomalous diffusion of system. Maike A. F. dos Santos, Marcelo K. Lenzi, and Ervin K. Lenzi Copyright © 2017 Maike A. F. dos Santos et al. All rights reserved. Canonical Forms and Their Integrability for Systems of Three 2nd-Order ODEs Sun, 16 Jul 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/2580968/ Differential invariants and their corresponding canonical forms for systems of three 2nd-order ODEs possessing three-dimensional Lie algebras are constructed. Their extension up to th-order system of three 2nd-order ODEs is presented. Furthermore singularity in invariant structure for the canonical forms is investigated. In addition integrability of these canonical forms is discussed. Illustrative physical examples from mechanics of system of particles are provided. S. Zahida, M. N. Qureshi, and Muhammad Ayub Copyright © 2017 S. Zahida et al. All rights reserved. Effect of Internal Heat Source on the Onset of Double-Diffusive Convection in a Rotating Nanofluid Layer with Feedback Control Strategy Thu, 13 Jul 2017 10:55:46 +0000 http://www.hindawi.com/journals/amp/2017/2789024/ A linear stability analysis has been carried out to examine the effect of internal heat source on the onset of Rayleigh–Bénard convection in a rotating nanofluid layer with double diffusive coefficients, namely, Soret and Dufour, in the presence of feedback control. The system is heated from below and the model used for the nanofluid layer incorporates the effects of thermophoresis and Brownian motion. Three types of bounding systems of the model have been considered which are as follows: both the lower and upper bounding surfaces are free, the lower is rigid and the upper is free, and both of them are rigid. The eigenvalue equations of the perturbed state were obtained from a normal mode analysis and solved using the Galerkin method. It is found that the effect of internal heat source and Soret parameter destabilizes the nanofluid layer system while increasing the Coriolis force, feedback control, and Dufour parameter helps to postpone the onset of convection. Elevating the modified density ratio hastens the instability in the system and there is no significant effect of modified particle density in a nanofluid system. I. K. Khalid, N. F. M. Mokhtar, I. Hashim, Z. B. Ibrahim, and S. S. A. Gani Copyright © 2017 I. K. Khalid et al. All rights reserved. Exact Partition Function for the Random Walk of an Electrostatic Field Thu, 13 Jul 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/6970870/ The partition function for the random walk of an electrostatic field produced by several static parallel infinite charged planes in which the charge distribution could be either is obtained. We find the electrostatic energy of the system and show that it can be analyzed through generalized Dyck paths. The relation between the electrostatic field and generalized Dyck paths allows us to sum overall possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example. Gabriel González Copyright © 2017 Gabriel González. All rights reserved. Intrinsic Optimal Control for Mechanical Systems on Lie Group Wed, 12 Jul 2017 07:47:57 +0000 http://www.hindawi.com/journals/amp/2017/6302430/ The intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the Hamilton-Jacobi-Bellman equation. For a special case on , the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance. Chao Liu, Shengjing Tang, and Jie Guo Copyright © 2017 Chao Liu et al. All rights reserved. Quasi-Particles, Thermodynamic Consistency, and the Gap Equation Mon, 10 Jul 2017 09:52:12 +0000 http://www.hindawi.com/journals/amp/2017/6265427/ The thermodynamic potentials of superconducting electrons are derived by means of the Bogoliubov-Valatin formalism. The thermodynamic potentials can be obtained by computing the free energy of a gas of quasi-particles, whose energy spectrum is conditional on the gap function. However, the nontrivial dependence of the gap on the temperature jeopardises the validity of the standard thermodynamic relations. In this article, it is shown how the thermodynamic consistency (i.e., the validity of the Maxwell relations) is recovered, and the correction terms to the quasi-particles potentials are computed. It is shown that the Bogoliubov-Valatin transformation avoids the problem of the thermodynamic consistency of the quasi-particle approach; in fact, the correct identification of the variables, which are associated with the quasi-particles, leads to a precise calculation of the quasi-particles vacuum energy and of the dependence of the chemical potential on the electron density. The stationarity condition for the grand potential coincides with the gap equation, which guarantees the thermodynamic consistency. The expressions of various thermodynamic potentials, as functions of the variables, are produced in the low temperature limit; as a final check, a rederivation of the condensation energy is presented. Enore Guadagnini Copyright © 2017 Enore Guadagnini. All rights reserved. Binormal Motion of Curves with Constant Torsion in 3-Spaces Thu, 29 Jun 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/7075831/ We study curve motion by the binormal flow with curvature and torsion depending velocity and sweeping out immersed surfaces. Using the Gauss-Codazzi equations, we obtain filaments evolving with constant torsion which arise from extremal curves of curvature energy functionals. They are “soliton” solutions in the sense that they evolve without changing shape. Josu Arroyo, Óscar J. Garay, and Álvaro Pámpano Copyright © 2017 Josu Arroyo et al. All rights reserved. Conversion of Monte Carlo Steps to Real Time for Grain Growth Simulation Tue, 27 Jun 2017 09:51:08 +0000 http://www.hindawi.com/journals/amp/2017/4023470/ Monte Carlo (MC) technique is becoming a very effective simulation method for prediction and analysis of the grain growth kinetics at mesoscopic level. It should be noted that MC models have no real time of physical systems due to the probabilistic nature of this simulation technique. This leads to difficulties when converting simulated time, the Monte Carlo steps , to real time. The correspondence between Monte Carlo steps and real time should be proposed for comparing the kinetics of MC models with the experiments. In this work, the conversion of Monte Carlo steps to real time is attempted. The lattice sites spacing Δ and the temperature cannot be ignored in the Monte Carlo simulation of grain growth. Real time will be associated with , , and Δ. N. Maazi Copyright © 2017 N. Maazi. All rights reserved. Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions Tue, 27 Jun 2017 09:21:07 +0000 http://www.hindawi.com/journals/amp/2017/5196513/ Given a bounded domain with a Lipschitz boundary and , we consider the quasilinear elliptic equation in complemented with the generalized Wentzell-Robin type boundary conditions of the form on . In the first part of the article, we give necessary and sufficient conditions in terms of the given functions , and the nonlinearities , , for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of “nonlinear Fredholm alternative” for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an -setting. Ciprian G. Gal and Mahamadi Warma Copyright © 2017 Ciprian G. Gal and Mahamadi Warma. All rights reserved. Heat Transfer in a Porous Radial Fin: Analysis of Numerically Obtained Solutions Tue, 27 Jun 2017 07:59:39 +0000 http://www.hindawi.com/journals/amp/2017/1658305/ A time dependent nonlinear partial differential equation modelling heat transfer in a porous radial fin is considered. The Differential Transformation Method is employed in order to account for the steady state case. These solutions are then used as a means of assessing the validity of the numerical solutions obtained via the Crank-Nicolson finite difference method. In order to engage in the stability of this scheme we conduct a stability and dynamical systems analysis. These provide us with an assessment of the impact of the nonlinear sink terms on the stability of the numerical scheme employed and on the dynamics of the solutions. R. Jooma and C. Harley Copyright © 2017 R. Jooma and C. Harley. All rights reserved. Conditional Well-Posedness for an Inverse Source Problem in the Diffusion Equation Using the Variational Adjoint Method Tue, 27 Jun 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/6801260/ This article deals with an inverse problem of determining a linear source term in the multidimensional diffusion equation using the variational adjoint method. A variational identity connecting the known data with the unknown is established based on an adjoint problem, and a conditional uniqueness for the inverse source problem is proved by the approximate controllability to the adjoint problem under the condition that the unknowns can keep orders locally. Furthermore, a bilinear form is set forth also based on the variational identity and then a norm for the unknowns is well-defined by which a conditional Lipschitz stability is established. Chunlong Sun, Qian Liu, and Gongsheng Li Copyright © 2017 Chunlong Sun et al. All rights reserved. Izergin-Korepin Analysis on the Projected Wavefunctions of the Generalized Free-Fermion Model Tue, 20 Jun 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/7563781/ We apply the Izergin-Korepin analysis to the study of the projected wavefunctions of the generalized free-fermion model. We introduce a generalization of the -operator of the six-vertex model by Bump-Brubaker-Friedberg and Bump-McNamara-Nakasuji. We make the Izergin-Korepin analysis to characterize the projected wavefunctions and show that they can be expressed as a product of factors and certain symmetric functions which generalizes the factorial Schur functions. This result can be seen as a generalization of the Tokuyama formula for the factorial Schur functions. Kohei Motegi Copyright © 2017 Kohei Motegi. All rights reserved. Quantum Cosmology of Quadratic Theories with a FRW Metric Sun, 18 Jun 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/1056514/ We study the quantum cosmology of a quadratic theory with a FRW metric, via one of its equivalent Horndeski type actions, where the dynamic of the scalar field is induced. The classical equations of motion and the Wheeler-DeWitt equation, in their exact versions, are solved numerically. There is a free parameter in the action from which two cases follow: inflation + exit and inflation alone. The numerical solution of the Wheeler-DeWitt equation depends strongly on the boundary conditions, which can be chosen so that the resulting wave function of the universe is normalizable and consistent with Hermitian operators. V. Vázquez-Báez and C. Ramírez Copyright © 2017 V. Vázquez-Báez and C. Ramírez. All rights reserved. Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with Sun, 04 Jun 2017 07:14:24 +0000 http://www.hindawi.com/journals/amp/2017/9743475/ Based on the three-dimensional real special orthogonal Lie algebra , by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities. Jian Zhang, Chiping Zhang, and Yunan Cui Copyright © 2017 Jian Zhang et al. All rights reserved.