Stationary and nonstationary nonlinear dynamics of the finite sine-lattice

Abstract

We present the results of the analytical as well as numerical study of the stationary and nonstationary dynamics of the sine-lattice. The latter is the discrete constitutive model used in various fields of physics, in particular, for the description of flexible polymers, quasi-one-dimensional spin chains, biopolymers, etc. To analyze the sine-lattice dynamics, we introduce the complex functions that allow us to determine the nonlinear normal modes as the stationary solutions to the equations in the wide range of the oscillation amplitudes and the wavenumbers. We present the dispersion relations in the analytical form. Analysis of the slow nonstationary processes allows us to determine the conditions of energy localization in the chain. We observe a good agreement between the analytical and numerical values of the localization thresholds for the chains of different lengths. In the long-wavelength approximation, the sine-lattice is equivalent to the Frenkel–Kontorova model. We demonstrate in the continuum limit that the equation is reduced to the nonlinear Schrödinger equation instead of the well-known sine-Gordon equation. We reveal the conditions of the existence of a breather-like solution and reduce the analytical representation for the small-amplitude approximation. We consider nonstationary dynamics of the forced oscillations for the undamped system in terms of the limiting phase trajectory; its bifurcations determine the change of the oscillatory regimes. We discuss the effect of damping on the slow system dynamics. We also present the generalized equation for the stationary amplitude of the forced oscillations in the presence of the damping.

Appendices

Appendix A: Multi-scale expansion

Substituting expression (7) into Eq. (6) and multiplying the latter by factor $e^{i \omega \tau _0}$, we should keep the terms of lower orders of a small parameter. In such a case, we obtain:

$$\begin{aligned}&i\frac{\partial \psi _{j,0}}{\partial \tau _0}+i \varepsilon \frac{\partial \psi _{j,0}}{\partial \tau _1}+i \varepsilon \frac{\partial \psi _{j,1}}{\partial \tau _0}+ \varepsilon \mu e^{i \omega \tau _0 }\nonumber \\&\quad \Biggl ( \frac{\omega }{2} \left( \psi _{j,0} e^{-i \omega \tau _0 } + cc \right) + \sum _{k=0}^{\infty } \frac{\left( 2 \omega \right) ^{-(k+1)}}{(2 k+1)!}\nonumber \\&\quad \biggl [ \beta \biggl ( \left( \left( \psi _{j+1,0}-\psi _{j,0} \right) e^{-i \omega \tau _0 } + cc \right) ^{2 k+1} \nonumber \\&\quad -\left( \left( \psi _{j,0}-\psi _{j-1,0} \right) e^{-i \omega \tau _0 } + cc \right) ^{2 k+1} \biggr )\nonumber \\&\quad - \sigma \left( \psi _{j,0} e^{-i \omega \tau _0 } + cc \right) ^{2 k+1} \biggr ] \Biggr )=0. \end{aligned}$$

(A.1)

Separating the terms with different orders of $\varepsilon$, we can write in zeroth order:

$$\begin{aligned} \varepsilon ^0: \; i\frac{\partial \psi _{j,0}}{\partial \tau _0}=0. \end{aligned}$$

(A.2)

Thus, function $\psi _{j,0}$ does not depend on the fast time $\tau _0$.

$\varepsilon ^1:$

$$\begin{aligned}&i\frac{\partial \psi _{j,1}}{\partial \tau _0}+ i\frac{\partial \psi _{j,0}}{\partial \tau _1}+ \mu \frac{\omega }{2} \psi _{j,0} + \mu \sum _{k=0}^{\infty } \frac{\left( 2 \omega \right) ^{-(k+1)}}{k! (k+1)!}\nonumber \\&\quad \biggl (\beta \biggl (| \psi _{j+1,0}-\psi _{j,0}|^{2k} \left( \psi _{j+1,0}-\psi _{j,0}\right) \nonumber \\&\quad - | \psi _{j,0}-\psi _{j-1,0}|^{2k} \left( \psi _{j,0}-\psi _{j-1,0}\right) \biggr )\nonumber \\&\quad - \sigma | \psi _{j,0}|^{2k} \psi _{j,0}\biggr )\nonumber \\&\quad + \mu \frac{\omega }{2} \psi ^*_{j,0} e^{2 i \tau _0 \omega } + \mu \sum _{k=0}^{\infty } \sum _{m \ne k}^{2k+1}\frac{\left( 2 \omega \right) ^{-(k+1)}}{m! (2 k-m+1)!} \nonumber \\&\quad \biggl ( \beta \left( \left( \psi ^{*}_{j+1,0}-\psi ^{*}_{j,0}\right) ^m \left( \psi _{j+1,0}-\psi _{j,0}\right) ^{2 k-m+1} \right. \nonumber \\&\quad \left. -\left( \psi ^*{}_{j,0}-\psi ^{*}_{j-1,0}\right) {}^m \left( \psi _{j,0}-\psi _{j-1,0}\right) {}^{2 k-m+1} \right) \nonumber \\&\quad -\sigma \psi _{j,0}^{2 k+1} \psi _{j,0} ^{*m} \biggr ) e^{-i 2 \tau _0 \omega (k-m+1)}=0. \end{aligned}$$

(A.3)

After integrating over fast time $\tau _0$, the fast oscillating terms vanish. The terms that contain the main order function $\psi _{j,0}$ do not depend on the fast time, and integrating leads to the secular term. Performing the summation, we get Eq. (8). So, we conclude that

$$\begin{aligned} i\frac{\partial \psi _{j,1}}{\partial \tau _0}=0. \end{aligned}$$

(A.4)

Appendix B: Integral of the occupation numbers

Using Eq. (13), we can receive evidence that the occupation number X is the integral of motion in the slow timescale:

$$\begin{aligned} i \frac{\partial X}{\partial \tau }= & {} i \sum _j{\left( \psi _j^*\frac{\partial \psi _j}{\partial \tau }+\psi _j \frac{\partial \psi _j^*}{\partial \tau } \right) }\nonumber \\= & {} \sum _j{\left( \psi _j^* \frac{\partial H}{\partial \psi _j^*}-\psi _j \frac{\partial H}{\partial \psi _j} \right) } \nonumber \\= & {} \sum _j{\left( \psi _j^* \frac{\partial H}{\partial X} \frac{\partial X}{\partial \psi _j^*}-\psi _j \frac{\partial H}{\partial X} \frac{\partial X}{\partial \psi _j} \right) }\nonumber \\= & {} \sum _j{\frac{\partial H}{\partial X} \left( \psi _j^* \psi _j-\psi _j \psi _j^* \right) } =0. \end{aligned}$$

(B.1)

It is easy to show that transformation (16) preserves the occupation number in the form (18):

$$\begin{aligned} X= & {} \sum _j{|\psi _j|^2}=\frac{1}{N}\sum _j \bigl ( \frac{1}{2}|\chi _1+\chi _2|^2\nonumber \\&+|\chi _1-\chi _2|^2\cos ^2{\left( \kappa j+\frac{\pi }{4}\right) }\nonumber \\&+ \left( \cos {\kappa j}-\sin {\kappa j} \right) |\chi _1-\chi _2|^2 \bigr )\nonumber \\= & {} \frac{1}{2} \left( |\chi _1+\chi _2|^2+|\chi _1-\chi _2|^2 \right) \nonumber \\= & {} |\chi _1|^2+|\chi _2|^2 \quad . \end{aligned}$$

(B.2)