Quasi-One-Dimensional Nonlinear Lattices

Abstract

In this section, it is shown how the LPT concept can be extended to finite-dimensional oscillatory chains. The systems under consideration are finite-dimensional analogues of several classical infinite models which were initially used for analysis of such significant physical phenomena as recurrent energy transfer and localization.

Appendix

Let us show that the pair of modes ${\mathbf{V}}_{0i} ,{\mathbf{V}}_{\varepsilon i}$ is orthogonal

$$ {\mathbf{V}}_{0i}^{\text{T}} {\mathbf{V}}_{\varepsilon i} = 0 $$

(4.97)

From the consideration of symmetry, the modes have the following form

$$ \begin{aligned} {\mathbf{V}}_{0i} & = \left[ {\begin{array}{*{20}c} 0 & {X_{2}^{0} } & \cdots & {X_{N + 1}^{0} } & 0 & {( - 1)^{r(i)} X_{2}^{0} } & \cdots & {( - 1)^{r(i)} X_{N + 1}^{0} } \\ \end{array} } \right] \\ {\mathbf{V}}_{\varepsilon i} & = \left[ {\begin{array}{*{20}c} {X_{1}^{\varepsilon } } & {X_{2}^{\varepsilon } } & \cdots & {X_{N + 1}^{\varepsilon } } & {( - 1)^{p(i)} X_{1}^{\varepsilon } } & {( - 1)^{p(i)} X_{2}^{\varepsilon } } & \cdots & {( - 1)^{p(i)} X_{N + 1}^{\varepsilon } ]} \\ \end{array} } \right] \\ \end{aligned} $$

(4.98)

Using perturbation theory, we have shown that

$$ {\text{X}}_{j}^{\varepsilon } = {\text{X}}_{j}^{0} + O(\varepsilon ) $$

(4.99)

where $X_{j}^{0} \sim\,O\left( 1 \right).$

Therefore,

$$ \text{sgn} ({X}_{j}^{\varepsilon } ) = \text{sgn} ({X}_{j}^{0} ),j = 2, \quad \dots N + 1 $$

(4.100)

Note that (A4) can be violated when dealing with the mode possessing the nodal points on some of the light particles (i.e., ${X}_{j}^{0} = 0,j \in [2,\dots, N]$). However, in this case, the product ${X}_{j}^{0} {X}_{j}^{\varepsilon } = 0$ and therefore the disparity in signs do not affect the orthogonality of the modes.

For each mode ${\mathbf{V}}_{0i}$, we note the two possibilities for the choice of $r(i)$. To this end, let us consider the signs of ${X}_{2}^{0}$ and ${X}_{N + 1}^{0}$. In case ${X}_{2}^{0}$ and ${X}_{N + 1}^{0}$ have identical signs, the immobile, heavy masses can be balanced by the oscillating light neighbors only if $r(i) = 1$.

Thus,

$$ r(i) = \left\{ {\begin{array}{*{20}c} {1,} & {\text{sgn} \left( {{\text{X}}_{2}^{0} } \right) = \text{sgn} \left( {{X}_{N + 1}^{0} } \right)} \\ {0,} & {\text{sgn} \left( {{X}_{2}^{0} } \right) = - \text{sgn} \left( {{X}_{N + 1}^{0} } \right)} \\ \end{array} } \right. $$

(4.101)

As for the second mode ${\mathbf{V}}_{\varepsilon i}$, it is obvious that operating on the optical branch of the chain, each oscillating heavy particle should be out-of-phase with its light neighbors. In other words, both the light particles neighboring to any of heavy masses should move in phase. Thus,

$$ p(i) = \left\{ {\begin{array}{*{20}l} {1,} & {\text{sgn} \left( {{X}_{2}^{\varepsilon } } \right) = \text{sgn} \left( {{X}_{N + 1}^{\varepsilon } } \right)} \\ {0,} & {\text{sgn} \left( {{X}_{2}^{\varepsilon } } \right) = - \text{sgn} \left( {{X}_{N + 1}^{\varepsilon } } \right)} \\ \end{array} } \right. $$

(4.102)

From (4.100)–(4.102), it is evident that

$$ r(i) \ne p(i) $$

(4.103)

Thus, the motion of particles for both the modes ${\mathbf{V}}_{0i} ,{\mathbf{V}}_{\varepsilon i}$ is in-phase on one cell of the dimer chain and out-of-phase on the second one. Accounting for (4.98) and (4.103), it is easy to see that

$$ {\mathbf{V}}_{0i}^{\text{T}} {\mathbf{V}}_{\varepsilon i} = 0 $$

(4.104)