Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

光测弹性理论中的耦联变分原理和广义耦联变分原理

在本文中,应用拉格朗日乘子法和高阶拉格朗日乘子法[1],我们系统地导出了光测弹性理论中的耦联势能原理,耦联余能原理和具有二类和三类变量的广义耦联势能原理和广义相联余能原理。     

Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation

In this paper, the prolongation structures of a generalized coupled Korteweg-de Vries (KdV) equation are investigated and two integrable coupled KdV equations associated with their Lax pairs are derived. Furthermore, a Miura transformation related to a integrable coupled KdV equation is derived, from which a new coupled modified KdV equations is obtained.     

Dynamics of coupled systems

A survey in the form of a review of the author’s research results in the area of dynamics of coupled rotations and coupled systems is presented. The theorem on the existence of a trigger of the coupled singularities and the separatrix in the form of the number eight is presented with a few examples of coupled rotations of the simple systems with debalances. Also, a survey of models and dynamics of coupled systems composed of a number of deformable bodies (plates, beams or belts) with different properties of materials and discrete layer properties is presented and mathematically described. The constitutive stress–strain relations for materials of the coupled sandwich structure elements are presented for different properties: elastic, viscoelastic and creeping. The characteristic modes of the coupled system vibrations are obtained and analyzed for different kinds of materials and structure composition. The visualization of the characteristic numbers and modes and eigenamplitude forms are presented. Structural analysis of sandwich structure vibrations is done.     

Multi-displacement continuum modelling of the metamaterial plate with periodical arranged resonators

Multi-displacement continuum modelling of a two-dimensional (2D) elastic metamaterials plate with periodically arranged local resonator over the surface of the plate is studied in this paper. The additional displacement fields are introduced to model the response of the local resonators. The continuous conditions between the adjacent unit-cells are used to reflect the periodicity of the microstructured continuum and resultantly turned into the constraint conditions between the additional displacement field and the other continuous field. The dispersion features of the multiple-displacement coupled wave propagating along the high symmetrical direction and any oblique direction are both studied numerically. It is found that the multi-displacement coupled waves can be divided into the coupled longitudinal wave and the coupled transversal wave when propagating along the highly symmetric direction but cannot be divided into the coupled longitudinal wave and the coupled transversal wave when propagating along any oblique direction. The effects of boundary conditions on the dispersion of acoustic and optical branches of coupled waves are discussed in detail. At last, the influences of the parameters of resonator on the dispersion feature of the multi-displacement coupled waves are investigated numerically.     

Ergodic dynamics of the coupled quasigeostrophic-flow-energy-balance system

The authors consider a mathematical model for the coupled atmosphere-ocean system, namely, the coupled quasigeostrophic-flow-energy-balance model. This model consists of the large-scale quasigeostrophic oceanic flow model and the transport equation for oceanic temperature, coupled with an atmospheric energy-balance model. After reformulating this coupled model as a random dynamical system (cocycle property), it is shown that the coupled quasigeostrophic-energy balance fluid system has a random attractor, and under further conditions on the physical data and the covariance of the noise, the system is ergodic, namely, for any observable of the coupled atmosphere-ocean flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 67–84, 2006.     

Mode decomposition for a synchronous state and its applications

Synchronization of coupled dynamical systems including periodic and chaotic systems is investigated both anlaytically and numerically. A novel method, mode decomposition, of treating the stability of a synchronous state is proposed based on the Floquet theory. A rigorous criterion is then derived, which can be applied to arbitrary coupled systems. Two typical numerical examples: coupled Van der Pol systems (corresponding to the case of coupled periodic oscillators) and coupled Lorenz systems (corresponding to the case of chaotic systems) are used to demonstrate the theoretical analysis.     

Dynamical Locality of the Nonminimally Coupled Scalar Field and Enlarged Algebra of Wick Polynomials

We discuss dynamical locality in two locally covariant quantum field theories: the nonminimally coupled scalar field and the enlarged algebra of Wick polynomials. We calculate the relative Cauchy evolution of the enlarged algebra, before demonstrating that dynamical locality holds in the nonminimally coupled scalar field theory. We also establish dynamical locality in the enlarged algebra for the minimally coupled massive case and the conformally coupled massive case.     

Integrability of the coupled KdV equations derived from two-layer fluids: Prolongation structures and Miura transformations

The Lax integrability of the coupled KdV equations derived from two-layer fluids [S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, Coupled KdV equations derived from two-layer fluids, J. Phys. A: Math. Gen. 39 (2006) 513-527] is investigated by means of prolongation technique. As a result, the Lax pairs of some Painlevé integrable coupled KdV equations and several new coupled KdV equations are obtained. Finally, the Miura transformations and some coupled modified KdV equations associated with the Lax integrable coupled KdV equations are derived by an easy way.     

Linearly and nonlinearly bidirectionally coupled synchronization of hyperchaotic systems

To date, there have been many results about unidirectionally coupled synchronization of chaotic systems. However, much less work is reported on bidirectionally-coupled synchronization. In this paper, we investigate the synchronization of two bidirectionally coupled Chen hyperchaotic systems, which are coupled linearly and nonlinearly respectively. Firstly, linearly coupled synchronization of two hyperchaotic Chen systems is investigated, and a theorem on how to choose the coupling coefficients are developed to guarantee the global asymptotical synchronization of two coupled hyperchaotic systems. Analysis shows that the choice of the coupling coefficients relies on the bound of the chaotic system. Secondly, the nonlinearly coupled synchronization is studied; a sufficient condition for the locally asymptotical synchronization is derived, which is independent of the bound of the hyperchaotic system. Finally, numerical simulations are included to verify the effectiveness and feasibility of the developed theorems.     

一类混合单调算子的耦合不动点定理及其应用

讨论了一类混合单调算子的耦合不动点定理,并获得了最大最小耦合不动点.作为应用,讨论了Banach空间中含有不连续项的混合单调Volterra型积分方程耦合拟解的存在性问题.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Integrability and singularity structure of coupled nonlinear Schrödinger equations

Multimode propagation of electromagnetic waves in optical fibre is often described by coupled nonlinear Schrödinger (NLS) equations. To understand the integrability properties of such coupled NLS systems, we extend the Painlevé singularity structure analysis of two coupled systems to three coupled systems and identify four integrable sets of parameters. We bilinearize these cases to obtain soliton solutions. The results are extended to N-coupled systems, completing the earlier analysis of Sahadevan, Tamizhmani and Lakshmanan.     

Invertible Coupled KdV and Coupled Harry Dym Hierarchies

In this paper, we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one‐forms, recursion operators, Poisson and symplectic operators. We show that the invertible coupled KdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.     

Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations

In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.     

The boundary integral method for the linearized rotating Navier-Stokes equations in exterior domain

In this paper, we apply the boundary integral method to the linearized rotating Navier-Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and an infinite domain, we obtain a coupled problem by the linearized rotating Navier-Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence and uniqueness of solution. Finally, we study the finite element approximation for the coupled problem and obtain the error estimate between the solution of the coupled problem and its approximation solution.     

New exact solutions for some coupled nonlinear partial differential equations using extended coupled sub-equations expansion method

An extended coupled sub-equations expansion method is proposed to seek new traveling wave solutions of the coupled nonlinear partial differential equations. The generalized Schrödinger-Boussinesq and coupled nonlinear Klein-Gordon-Schrödinger equations are used to illustrate the validity and the advantages of this method.     

Effect of micro-inertia in the propagation of waves in micropolar thermoelastic materials with voids

The effect of micro-inertia in the propagation of waves in micropolar thermoelastic materials with voids has been investigated. Elastic waves are reflected due to incident coupled longitudinal and coupled shear waves from a plane free boundary of micropolar thermoelastic materials with voids. The amplitude ratios corresponding to the reflected coupled longitudinal and coupled shear waves are derived by using appropriate boundary conditions. Energy partition in the free surface has been presented. The amplitude and energy ratios of the reflected waves are also computed numerically for a particular model.     

On the problem of propagation of nonlinear coupled TE-TM waves in a layer

The problem of simultaneous propagation of two types of electromagnetic waves (TE and TM) in a plane dielectric waveguide filled with a nonlinear medium is considered. These polarized waves have different frequencies and different propagation constants. The physical problem is reduced to a nonlinear two-parameter transmission eigenvalue problem for Maxwell’s equations in a layer. The coupled eigenvalues are coupled propagation constants. A theorem on the existence and localization of coupled eigenvalues corresponding to coupled polarized electromagnetic waves is proved.     

Universal occurrence of mixed-synchronization in counter-rotating nonlinear coupled oscillators

By coupling counter-rotating coupled nonlinear oscillators, we observe a “mixed” synchronization between the different dynamical variables of the same system. The phenomenon of amplitude death is also observed. Results for coupled systems with co-rotating coupled oscillators are also presented for a detailed comparison. Results for Landau–Stuart and Rössler oscillators are presented.