A note on “A study on an integrable system of coupled KdV equations”

We analyze the paper by Wazwaz [Wazwaz AM. A study on an integrable system of coupled KdV equations. Commun Nonlinear Sci Numer Simul, 2010;15:2846–2850]. Author tried to show that the system of coupled KdV equations is completely integrable but he has used the curious approach for the proof. We demonstrate that, author has taken the relation between dependent variables and has obtained the well known result by Hirota for the KdV equation.     

A note on solutions of the Korteweg–de Vries hierarchy

olutions of the Korteweg–de Vries hierarchy are discussed. It is shown that results by Wazwaz [Wazwaz AM. Multiple-soliton solutions of the perturbed KdV equation. Commun Nonlinear Sci Simul 2010;15911:3270–73] are the well-known consequences of the full integrability for the Korteweg–de Vries hierarchy.     

A study of the solutions of the combined sine–cosine-Gordon equation

We have studied the solutions of the combined sine–cosine-Gordon Equation found by Wazwaz [A.M. Wazwaz, Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method, Appl. Math. Comput. 177 (2006) 755] using the variable separated ODE method. These solutions can be transformed into a new form. We have derived the relation between the phase of the combined sine–cosine-Gordon equation and the parameter in these solutions. Its applications in physical systems are also discussed.     

A note on “The integrable KdV6 equation: Multiple soliton solutions and multiple singular soliton solutions”

The goal of this short note is to provide another kind soliton solutions with Hirota form, which is different from what Wazwaz obtained in [A.M. Wazwaz, The integrable KdV6 equations: Multiple soliton solutions and multiple singular soliton solutions, Appl. Math. Comput. 204 (2008) 963-972]. Meanwhile we newly construct the MKdV6 equation and derive a Miura transformation between KdV6 equation and MKdV6 equation.     

Redundant exact solutions of nonlinear differential equations

We analyze the paper by Wazwaz and Mehanna [Wazwaz AM, Mehanna MS. A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation. Appl Math Comput 2010;217:1484–90]. The authors claim that they have found exact solutions of the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation using the tanh–coth method and the Exp-function method. We demonstrate that two of their solutions are incorrect. All the others can be simplified and they are the partial cases of the well-known solution. Wazwaz and Mehanna made a number of typical mistakes in finding exact solutions of nonlinear differential equations. Taking the results of this paper we introduce the definition of redundant exact solutions for the nonlinear ordinary differential equations.     

Classification, reduction, group invariant solutions and conservation laws of the Gardner-KP equation

In this paper, symmetries and group invariant solutions to the Gardner-KP equation are obtained by using the direct symmetry method. At the same time, we find the corresponding Lie algebra, optimal system, classification and the similarity reductions to the equation, respectively. Our exact solutions generalize the corresponding results obtained by Wazwaz. In addition, the conservation laws of Gardner-KP equation are also given.     

(2+1)维AKNS方程的行波解研究(英文)

Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic solution. It also proves that the result is consistent with the soliton solution of simplify Hirota bilinear method by Wazwaz and illustrate the solution are right travelling wave solution.     

Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation

By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. We notice that some results in [Wazwaz AM. Analytic study on nonlinear variants of the RLW and the PHI-four equations. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2005.03.001] are incorrect.     

Numerical solution of special 12th-order boundary value problems using differential transform method

In this paper, a differential transform method (DTM) is used to find the numerical solution of a special 12th-order boundary value problems with two point boundary conditions. The analysis is accompanied by testing differential transform method both on linear and nonlinear problems from the literature [Wazwaz AM. Approximate solutions to boundary value problems of higher-order by the modified decomposition method. Comput Math Appl 2000:40;679–91; Siddiqi SS, Ghazala Akram. Solutions of 12th order boundary value problems using non-polynomial spline technique. Appl Math Comput 2007. doi:10.1016/j.amc.2007.10.015; Siddiqi SS, Twizell EH. Spline solutions of linear 12th-order boundary value problems. J Comput Appl Math 1997;78:371–90]. Numerical experiments and comparison with existing methods are performed to demonstrate reliability and efficiency of the proposed method.     

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.     

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.     

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.     

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。     

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.     

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.     

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.     

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.     

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.     

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.