Wave Propagation and Resonance in a One-Dimensional Nonlinear Discrete Periodic Medium
Wave Propagation and Resonance in a One-Dimensional Nonlinear Discrete Periodic Medium
复制标题
DOI：
10.1137/s0036139998340315
复制
发表时间：
1999-11
期刊：
影响因子：
--
通讯作者：
Anna Georgieva;T. Kriecherbauer;S. Venakides
中科院分区：
文献类型：
--
作者：
Anna Georgieva;T. Kriecherbauer;S. Venakides
We consider wave propagation in a nonlinear infinite diatomic chain of particles as a discrete model of propagation in a medium whose properties vary periodically in space. The particles have alternating masses M1 and M2 and interact in accordance to a general nonlinear force F acting between the nearest neighbors. Their motion is described by the system of equations \begin{align*} \ddot{y_{n}} & = \frac{1}{M_{1}}(F(y_{n-1}-y_{n})- F(y_{n}-y_{n+1})),\\ \ddot{y}_{n+1} & = \frac{1}{M_{2}}(F(y_{n}-y_{n+1})-F(y_{n+1}- y_{n+2})), \end{align*} where $\{y_{n} \}_{n= -\infty}^{\infty}$ is the position of the nth particle.Using Fourier series methods and tools from bifurcation theory, we show that, for nonresonant wave-numbers k, this system admits nontrivial small-amplitude traveling wave solutions g and h, depending only on the linear combination $z=kn-\omega t$. We determine the nonlinear dispersion relation. We also show that the system sustains binary oscillations with arbitrarily large amplitude.
We consider wave propagation in a nonlinear infinite diatomic chain of particles as a discrete model of propagation in a medium whose properties vary periodically in space. The particles have alternating masses M1 and M2 and interact in accordance to a general nonlinear force F acting between the nearest neighbors. Their motion is described by the system of equations \begin{align*} \ddot{y_{n}} & = \frac{1}{M_{1}}(F(y_{n-1}-y_{n})- F(y_{n}-y_{n+1})),\\ \ddot{y}_{n+1} & = \frac{1}{M_{2}}(F(y_{n}-y_{n+1})-F(y_{n+1}- y_{n+2})), \end{align*} where $\{y_{n} \}_{n= -\infty}^{\infty}$ is the position of the nth particle.Using Fourier series methods and tools from bifurcation theory, we show that, for nonresonant wave-numbers k, this system admits nontrivial small-amplitude traveling wave solutions g and h, depending only on the linear combination $z=kn-\omega t$. We determine the nonlinear dispersion relation. We also show that the system sustains binary oscillations with arbitrarily large amplitude.