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(1. School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China;2. School of Public Administration, Guangxi Technological College of Machinery and Electrcity,Nanning 530007, China)

Abstract: In this paper, we study the periodic wave propagation phenomenon in elastic waveguides modeled by a combined double-dispersive partial differential equation (PDE).The traveling wave ansazt transforms the PDE model into a perturbed integrable ordinary differential equation (ODE). The global bifurcation theory is applied for the perturbed ODE model to establish the existence and uniqueness of the limit cycle, which corresponds the periodic traveling wave for the PDE model. The main tool is the Abelian integral taken from Poincar′e bifurcation theory. Simulation is carried out to verify the theoretical result.

Keywords: Combined double-dispersive PDE;Abelian integral;Limit cycle;Hamiltonian function; Periodic wave

§1. Introduction

In material science, one important mathematical study is to analyze the strain wave propagation phenomenon. This kind of wave propagation transfers energy over long distances along elastic waveguide. The uncertain influences, including external type such as relatively weak influence of external medium on the surface of the waveguide and interior type such as the inhomogeneity of the material, may cause amplification of the strain wave leading to microcracks or eventually more serious breakdown of the waveguide. Therefore, dynamical study on the waves is of great importance in durability assessment of the material mechanics [3,10].Elastic rods can be used in various kinds of tools and machines and appear in many kinds of mechanic studies. Let consider an isotropic cylindrical compressible elastic rod embedded in an external medium subjected to Kerr’s viscoelastic contact [4]. Porubov and Velarde [6,7]show that bell-shaped solitary wave propagation by analyzing the mathematical model built based on the Cauchy-Green deformation tensor analysis and Hamilton’s principle with the necessary quantities such as stiffness of medium, viscocompressibility of the external medium,e.t.c considered. The reduced mathematical model is a dissipation modified double dispersive partial differential equation (DMDDE for short),

where theαis are constants and depend on the physical quantities. Experimental study has been carried out to verify the theoretical prediction [9]. It should be noted that the solitary wave propagation happens within the balance among the quadratic nonlinearity, dissipation and dispersion. It is interesting to point out that the kink-shaped wave can also propagate in such kind of elastic rod within a sustained different balance,among which an important and necessary factor is the cubic nonlinearity due to the more strongly nonlinear elastic material. Porubov and Velarde [8] obtained the following partial differential equation by the similar mechanical reduction when involving the cubic nonlinearity.

whereαis in this equation depend on the related physical quantities. The exact kink-shaped solitary wave solution and bounded periodic wave solution were obtained in [8]. When the dissipative effect is weak, i.e. the external medium is of little influence, the viscocompressibility coefficient is small, thenα2andα6are small, then one can rewriteα2=∈?α2,α6=∈?α6where,∈is small and ?α2and ?α6are bounded. Then the equation (1.2) becomes a perturbed combined double-dispersive equation,

Porubov and Velarde [8] also proved that the weakly perturbed equation can have a kink-shaped wave. However, they could not prove whether the periodic traveling wave can propagate. In this paper, we apply the dynamical system theory to establish the existence of periodic wave and the sustained balance. We show that the periodic wave persists when the ratio of two Abelian integrals is located in a certain interval.

This paper is organized as follows. In section 2,we transform the PDE(1.3)into a perturbed dynamical system and explain why the periodic traveling wave can be controlled by a linear combination of two Abelian integral. We show the ratio of the two Abelian integrals is monotonic,for proving the first order approximation of the return map has at most one zero. This is the key step to establish the global existence and uniqueness of periodic wave for the weakly dissipative perturbation of the equation (1.3). Finally, we conduct numerical study to verify our theoretical result.

§2. Existence and uniqueness of a bounded periodic wave

In this section,we transform the PDE(1.3)into a perturbed integrable system and show that the periodic traveling wave can be controlled by a linear combination of two Abelian integral.We take the traveling wave anastzv(x,t)=ψ(η)=ψ(x-ct),cis the wave speed. Letψ(-∞)=a,ψ(+∞)=b,v(x,t) is a solitary wave solution ifa=b, and a kink or anti-kink solution ifa/=b.We substitutev(x,t)=ψ(η) into (1.3) and integrate the equation twice, we have

§3. Numerical simulations

Fig. 1 Simulated periodic solution of system (2.4) with β=, ∈=0.001, δ0=-0.3739896116,δ1=1 and the initial value (φ,z)=(,0).

Fig. 2 Simulated periodic solution of system (2.4) with β=, ∈=0.001, δ0=-0.3658612743,δ1=1 and the initial value (φ,z)=(,0).

Fig. 3 Simulated periodic solution of system (2.4) with β=, ∈=0.001, δ0=-0.3506924415,δ1=1 and the initial value (φ,z)=(,0).

§4. Conclusion

In this paper, we prove the weakly dissipative PDE modeling an elastic rotating rod can have periodic wave solution. The problem is reduced to a perturbed Hamiltonian system.By analyzing the monotonicity of the related two Abelian integrals, we prove the perturbed system can have an unique limit cycle, implying an unique periodic wave solution for the weakly dissipative model. The numerical study verifies the theoretical prediction.