Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method

In this paper, the Exp-function method with the aid of the symbolic computational system Maple is used to obtain the generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, (2+1)-dimensional Konopelchenko–Dubrovsky equations, the (3+1)-dimensional Jimbo–Miwa equation, the Kadomtsev–Petviashvili (KP) equation, and the (2+1)-dimensional sine-Gordon equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.     

Existence of time periodic solutions for a damped generalized coupled nonlinear wave equations

IntroductionInRef.[1 ] ,theauthorsestablishedtheuniqueexistenceofthesmoothsolutionforthefollowingcouplednonlinearequationsut=uxxx+buux+ 2vvx, (1 )vt=2 (uv) x. (2 )Thesewereproposedtodescribetheinteractionprocessofinternallongwaves.InRef.[2 ] ,ItoM .proposedarecursionoperatorbywhichheinferredthatEqs.(1 )and (2 )possesinfinitelymanysymmetriesandconstantsofmotion .InRef.[3 ] ,P .F .HeestablishedtheexistenceofasmoothsolutiontothesystemofcouplednonlinearKdVequation[4 ]ut=a(uxxx+buux) + 2bvvx,(…     

Travelling wave solutions for a second order wave equation of KdV type

The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type.In different regions of the parametric space,sufficient conditions to guarantee the existence of solitary wave solutions,periodic wave solutions,kink and anti-kink wave solutions are given.All possible exact explicit parametric representations are obtained for these waves.     

一种非线性系统周期解的延拓算法

提出了一种非线性系统周期解的延拓算法。指出了非线性系统周期解在分岔点处由于雅可比矩阵奇异而导致一般延拓方法延拓失败问题;然后基于推广的打靶法的思想,将普通延拓算法推广,提出了一种用于周期解延拓的算法。对于非线性动力系统,该算法可以在已知某一参数下的周期解的基础上,求解出在一定参数范围内非线性动力系统的解随参数的连续变化情况。应用该方法对非线性柔性转子-轴承系统的周期解与参数的依赖关系进行了求解,验证了方法的有效性。     

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M. Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.     

Further improved F-expansion and new exact solutions for nonlinear evolution equations

The improved F-expansion method with a computerized symbolic computation is used to construct the exact traveling wave solutions of four nonlinear evolution equations in physics. As a result, many exact traveling wave solutions are obtained which include new soliton-like solutions, trigonometric function solutions, and rational solutions. The method is straightforward and concise, and it holds promise for many applications.     

Existence of solutions for periodic boundary value problems for second-order integro-differential equations

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DISCUSSION ON THE PERIODIC SOLUTIONS FOR HIGHER-ORDER LINEAR EQUATION OF NEU

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THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

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Existence and nonexistence of periodic solutions for the Liénard and related equations

The existence of periodic solutions of Liénard type equations is converted into the existence of special fixed point problems with auxiliary conditions. A general method for exact calculation of the limit cycles is given, and the corresponding numerical iterative procedure is carried out in significant cases, with comparison to standard Runge-Kutta numerical integration. On the basis of the general theory, criteria for the existence of limit cycles are given and tested in particular cases.     

A new numerical technique for the analysis of parametrically excited nonlinear systems

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.     

MULTI-SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS AND KUPERSHMIDT EQUATIONS

Ｒｅｃｅｎｔｌｙ,ＷａｎｇｐｒｅｓｅｎｔｅｄａｈｏｍｏｇｅｎｅｏｕｓｂａｌａｎｃｅｄｍｅｔｈｏｄｆｏｒｃｏｎｓｔｒｕｃｔｉｎｇｓｏｌｉｔａｒｙｗａｖｅｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｖａｒｉａｎｔＢｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎｓ[1].Ｉｎｔｈｉｓｐａｐｅｒ,ｗｅｗｉｓｈｔｏｇｉｖｅｇｅｎｅｒａｌｍｕｌｔｉ＿ｓｏｌｉｔｏｎｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｖａｒｉａｎｔＢｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎｓａｎｄｔｈｅＫｕｐｅｒｓｈｍｉｄｔｅｑｕａｔｉｏｎｓ.Ｔｈｅｆｉｒｓｔｔｙｐｅｏｆｖａｒｉａｎｔ…     

On a new extended finite element method for dislocations: Core enrichment and nonlinear formulation

A recently developed finite element method for the modeling of dislocations is improved by adding enrichments in the neighborhood of the dislocation core. In this method, the dislocation is modeled by a line or surface of discontinuity in two or three dimensions. The method is applicable to nonlinear and anisotropic materials, large deformations, and complicated geometries. Two separate enrichments are considered: a discontinuous jump enrichment and a singular enrichment based on the closed-form, infinite-domain solutions for the dislocation core. Several examples are presented for dislocations constrained in layered materials in 2D and 3D to illustrate the applicability of the method to interface problems.     

A general reduction method for periodic solutions in conservative and reversible systems

We introduce a general reduction method for the study of periodic solutions near equilibria in autonomous systems which are either conservative or reversible. We impose no restrictions on the linearization at the equilibrium, allowing higher multiplicities and all kinds of resonances. It is shown that the problem reduces to a similar problem for a reduced system, which is itself conservative or reversible, but also has an additionalS ¹-symmetry. This symmetry allows to immediately write down the bifurcation equations. Moreover, the reduced system can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Liapunov-Schmidt method. A similar approach was already introduced for Hamiltonian systems in [9], and for equivariant systems in [3]; this paper extends the results of these papers to the cases of conservative and reversible systems.The research in this paper was supported by the EEC Science Project on Bifurcation Theory and Its Applications.     

On the construction of manufactured solutions for one and two‐equation eddy‐viscosity models

This paper presents manufactured solutions (MSs) for some well‐known eddy‐viscosity turbulence models, viz. the Spalart & Allmaras one‐equation model and the TNT and BSL versions of the two‐equation k–ω model. The manufactured flow solutions apply to two‐dimensional, steady, wall‐bounded, incompressible, turbulent flows. The two velocity components and the pressure are identical for all MSs, but various alternatives are considered for specifying the eddy‐viscosity and other turbulence quantities in the turbulence models. The results obtained for the proposed MSs with a second‐order accurate numerical method show that the MSs for turbulence quantities must be constructed carefully to avoid instabilities in the numerical solutions. This behaviour is model dependent: the performance of the Spalart & Allmaras and k–ω models is significantly affected by the type of MS. In one of the MSs tested, even the two versions of the k–ω model exhibit significant differences in the convergence properties. Copyright © 2006 John Wiley & Sons, Ltd.     

Time-harmonic Green's function and boundary integral formulation for incremental nonlinear elasticity: dynamics of wave patterns and shear bands

Superimposed dynamic, time-harmonic incremental deformations are considered in an elastic, orthotropic and incompressible, infinite body, subject to plane, homogeneous—but otherwise arbitrary—deformation. The dynamic, infinite body Green's function is found and, in addition, new boundary integral equations are obtained for incremental in-plane hydrostatic stress and displacements. These findings open the way to integral methods in incremental, dynamic elasticity. Moreover, the Green's function is employed as a dynamic perturbation to analyze interaction between wave propagation and shear band formation. Depending on anisotropy and pre-stress level, peculiar wave patterns emerge with focussing and shadowing effects of signals, which may remain undetected by the usual criteria based on analysis of weak discontinuity surfaces.