Perturbation and asymptotic solutions of energy localization of impurity modes in a one-dimensional anharmonic chain

A 1D anharmonic chain with a single impurity particle is used to study the vibrational energy localization. Numerical and asymptotic solutions for the symmetric anharmonic localized mode are both presented. The numerical results reveal that the energy localization strengthens with decreasing impurity mass or with increasing anharmonicity. In the weak-anharmonicity limit, the energy localization is close to the harmonic results and varies linearly with respect to the anharmonicity parameter. In the strong-anharmonicity limit, the localized structure tends towards a constant value that is independent of anharmonicity, but as a function of impurity mass. We finally analyze the stability of this symmetric impurity mode, and evidence a stable asymmetric mode as the result of a bifurcation from the symmetric mode for the case of the large impurity mass.

Figure 1

Numerical solutions (points) and perturbation solutions (lines) for (a) ${\varpi }^2$ and (b) $E_i/E_t$ at $\lambda =0.1,0.3$, and 0.6.

Figure 2

Numerical solutions (points) and perturbation solutions (lines) for (a) ${\varpi }^2$ and (b) $E_i/E_t$ at $f=0.1$.

Figure 3

Numerical solutions (points) and asymptotic solutions (lines) for (a) ${\varpi }^2$, and (b) $E_i/E_t$ at $\lambda =0.1,0.5$, and 0.9.

Figure 4

Numerical solutions (points) and asymptotic solutions (lines) for (a) ${\varpi }^2$, and (b) $E_i/E_t$ at $f=100$. The red curves are the asymptotic solutions, and the blue curves are the light-mass limit solutions.

Figure 5

(a) Logarithm of the square of the frequency, $\lg \left({\varpi }^2\right)$, and (b) $E_i/E_t$ vs λ and lg($f$).

Figure 6

(a) Bifurcation diagram for $\varpi =100$. The green and magenta branches show the λ characteristics of the symmetric modes (denoted by Sym. 1 and 2), and the black branch shows λ for the asymmetric mode (denoted by Asym.). (b) Total energies of both the symmetric and asymmetric modes for $\varpi =100$. The black branch is the asymmetric mode, and the red and blue branches are symmetric modes.

Figure 7

Critical value ${\lambda }_{\mathrm{cr}}$ vs frequency variable ε.