The monoatomic FPU system
as a limit of a diatomic FPU system

We consider a diatomic infinite Fermi–Pasta–Ulam (FPU) system depicted schematically on Figure 1. Displacements of heavy particles are denoted by $Q_{j}$ with $j\in 2\mathbb{Z}$, whereas displacements of light particles are denoted by $q_{j}$, with $j\in 2\mathbb{Z}+1$. For convenience, we normalize the mass of the heavy particles to unity and denote the mass ratio between masses of light and heavy particles by the parameter $\varepsilon^{2}$. The total energy of the diatomic system is

(1)

$$E=\sum_{j\in 2\mathbb{Z}}\frac{1}{2}\dot{Q}_{j}^{2}+\frac{1}{2}\varepsilon^{2}\dot{q}_{j+1}^{2}+W(q_{j+1}-Q_{j})+W(Q_{j}-q_{j-1}),$$

where the dot denotes the derivative in time $t$ and $W:\mathbb{R}\mapsto\mathbb{R}$ is a smooth potential for the pairwise interaction force between the adjacent light and heavy particles. Equations of motion are generated from the total energy (1) by using the standard symplectic structure for the dynamics of particles. They are written in the form:

(2)		$\displaystyle\ddot{Q}_{j}$	$\displaystyle=$	$\displaystyle W^{\prime}(q_{j+1}-Q_{j})-W^{\prime}(Q_{j}-q_{j-1}),$
(3)		$\displaystyle\varepsilon^{2}\ddot{q}_{j+1}$	$\displaystyle=$	$\displaystyle W^{\prime}(Q_{j+2}-q_{j+1})-W^{\prime}(q_{j+1}-Q_{j}),$

where $j\in 2\mathbb{Z}$.

The dynamics of diatomic lattices, e.g. propagation of traveling solitary waves, has always been important in physical applications and has been studied in numerous works, e.g. [3, 9, 21]. More recently, such diatomic systems were considered in the context of granular chains [13, 14, 18]. In particular, the authors of [13] proposed to consider the following reduction of the diatomic system in the limit of vanishing mass ratio $\varepsilon\to 0$:

$$0=W^{\prime}(Q_{j+2}-q_{j+1})-W^{\prime}(q_{j+1}-Q_{j})\quad\Rightarrow\quad Q_{j+2}-q_{j+1}=q_{j+1}-Q_{j},$$

which yields

(4)

$$q_{j+1}=\frac{Q_{j+2}+Q_{j}}{2}.$$

If $q_{j+1}$ is expressed by (4), the dynamics of the heavy particles is governed by the monoatomic FPU system:

(5)

$$\ddot{Q}_{j}=W^{\prime}\left(\frac{Q_{j+2}-Q_{j}}{2}\right)-W^{\prime}\left(\frac{Q_{j}-Q_{j-2}}{2}\right),$$

where $j\in 2\mathbb{Z}$. It follows from (4) and (5) that the light particles are squeezed by the heavy particles and move according to the mean values of their displacements, whereas the heavy particles move according to their pairwise interactions.

Numerical results on existence and non-existence of traveling solitary waves in the diatomic system (3) which are close to the traveling solitary waves of the monoatomic system (5) were reported in [13]. These numerical results inspired a number of analytical works where the authors developed the existence theory for traveling solitary waves with oscillatory tails [5, 8], beyond-all-order theory [16, 17], and the linearized analysis of perturbations [22]. It is the purpose of this paper to give rigorous error estimates for this approximation in the context of the initial-value problem.

Note that the small mass ratio limit for diatomic FPU system has been considered before in the context of the existence of breathers [11, 12, 15, 24] and traveling periodic waves [2, 10, 19, 20]. However, these works rely on the ideas of the so-called anti-continuum limit, for which the heavy particles do not move after rescaling of the time variable, whereas the light particles perform uncoupled oscillations in between the heavy particles. The limit (4) and (5) is clearly different from the anti-continuum limit.

Other relevant results on travelling solitary waves in diatomic lattices include persistence results near the equal mass ratio limit [4], asymptotic approximations near the long-wave limit [6, 23], and numerically assisted study of radiation generated from long-wave solitons in the time evolution [7].

We shall now present the main approximation theorem. We use the standard notation $\ell^{2}$ to denote square summable sequences equipped with the norm

$$\|u\|_{\ell^{2}}:=\left(\sum_{k\in\mathbb{Z}}|u_{k}|^{2}\right)^{1/2},$$

from which it is obvious that $\sup_{k\in\mathbb{Z}}|u_{k}|\leq\|u\|_{\ell^{2}}$. Another useful property of the $\ell^{2}$ space is being a Banach algebra with respect to pointwise multiplication.

The remainder of the paper is organized as follows. In Section 2, we rewrite the diatomic FPU system in new coordinates which are more suitable to express perturbations to the motion given by the limit system (4) and (5). The bounds in Theorem 1 are obtained with the energy estimates in Section 3 for the simple case with $W^{\prime}(u)=u+u^{2}$. Generalizations to other nonlinear interaction potentials $W(u)$ are discussed in Section 4.

Acknowledgement. G. Schneider is partially supported by the Deutsche Forschungsgemeinschaft DFG through the CRC 1173 “Wave phenomena”. D. Pelinovsky is partially supported by the grant of the President of the Russian Federation for scientific research of leading scientific schools of the Russian Federation NSh-2485.2020.5.

By using suitable chosen coordinates, we will separate the fast and slow dynamics of the diatomic FPU system (2)–(3) and will introduce perturbations to the motion given by the limit system (4)–(5). Note that the same choice of coordinates was used in [8] in the study of traveling waves. Let us set

$$U_{j}:=\frac{1}{2}(Q_{j+2}-Q_{j})\quad\textrm{and}\quad w_{j+1}:=q_{j+1}-\frac{1}{2}(Q_{j+2}+Q_{j}),$$

so that

$$q_{j+1}-Q_{j}=U_{j}+w_{j+1}\quad\textrm{and}\quad Q_{j+2}-q_{j+1}=U_{j}-w_{j+1}.$$

The diatomic FPU system (2)–(3) is now written as

$\displaystyle 2\ddot{U}_{j}$

$\displaystyle=$

$\displaystyle W^{\prime}(U_{j+2}+w_{j+3})-W^{\prime}(U_{j}-w_{j+1})-W^{\prime}(U_{j}+w_{j+1})+W^{\prime}(U_{j-2}-w_{j-1})$

and

	$\displaystyle\varepsilon^{2}\ddot{w}_{j+1}$	$\displaystyle=$	$\displaystyle W^{\prime}(U_{j}-w_{j+1})-W^{\prime}(U_{j}+w_{j+1})-\frac{1}{2}\varepsilon^{2}W^{\prime}(U_{j+2}+w_{j+3})$
			$\displaystyle+\frac{1}{2}\varepsilon^{2}W^{\prime}(U_{j}-w_{j+1})-\frac{1}{2}\varepsilon^{2}W^{\prime}(U_{j}+w_{j+1})+\frac{1}{2}\varepsilon^{2}W^{\prime}(U_{j-2}-w_{j-1}).$

For the particular choice $W^{\prime}(u)=u+u^{2}$ we obtain

	$\displaystyle W^{\prime}(U_{j}-w_{j+1})+W^{\prime}(U_{j}+w_{j+1})$	$\displaystyle=$	$\displaystyle 2U_{j}+2U_{j}^{2}+2w_{j+1}^{2},$
	$\displaystyle W^{\prime}(U_{j}-w_{j+1})-W^{\prime}(U_{j}+w_{j+1})$	$\displaystyle=$	$\displaystyle-2w_{j+1}-4U_{j}w_{j+1},$

which yields the following system of equations:

(9)		$\displaystyle\ddot{U}_{j}+U_{j}+U_{j}^{2}+w_{j+1}^{2}$	$\displaystyle=$	$\displaystyle g(U_{j+2},U_{j-2},w_{j+3},w_{j-1}),$
(10)		$\displaystyle\varepsilon^{2}\ddot{w}_{j+1}+(2+\varepsilon^{2})w_{j+1}(1+2U_{j})$	$\displaystyle=$	$\displaystyle\varepsilon^{2}h(U_{j+2},U_{j-2},w_{j+3},w_{j-1}),$

where

	$\displaystyle g(U_{j+2},U_{j-2},w_{j+3},w_{j-1})$	$\displaystyle=$	$\displaystyle\frac{1}{2}W^{\prime}(U_{j+2}+w_{j+3})+\frac{1}{2}W^{\prime}(U_{j-2}-w_{j-1}),$
	$\displaystyle h(U_{j+2},U_{j-2},w_{j+3},w_{j-1})$	$\displaystyle=$	$\displaystyle-\frac{1}{2}W^{\prime}(U_{j+2}+w_{j+3})+\frac{1}{2}W^{\prime}(U_{j-2}-w_{j-1}).$

The dynamics of $U$ and $w$ occurs now at two different scales: $U$ changes on the time scale of $\mathcal{O}(1)$, whereas $w$ changes on the faster time scale of $\mathcal{O}(\varepsilon)$. The approximation result of Theorem 1 justifies the dynamics of $U$ on the time scale of $\mathcal{O}(1)$. The dynamics of $w$ is slaved to the dynamics of $U$ on this time scale.

The leading-order approximation in the new coordinates is denoted by $(U,w)=(\Psi,0)$, where $\Psi$ satisfies

(11)

$$\ddot{\Psi}_{j}+\Psi_{j}+\Psi_{j}^{2}=g(\Psi_{j+2},\Psi_{j-2},0,0).$$

After inserting this approximation into the equations of motion, the remaining terms are collected in the residual, which is given by

	$\displaystyle\textrm{Res}_{U,j}$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle\textrm{Res}_{w,j}$	$\displaystyle=$	$\displaystyle\varepsilon^{2}h(\Psi_{j+2},\Psi_{j-2},0,0).$

Thanks for $\ell^{2}$ being a Banach algebra with respect to pointwise multiplication, the residual terms obey the following estimate.

For estimating the difference between the approximation and true solutions we introduce the error functions $R$ and $v$ by using the decomposition

(13)

$$U_{j}=\Psi_{j}+\varepsilon R_{j}\quad\textrm{and}\quad w_{j+1}=\varepsilon v_{j+1}.$$

These functions satisfy the following system

	$\displaystyle\ddot{R}_{j}+R_{j}+2\Psi_{j}R_{j}+\varepsilon R_{j}^{2}+\varepsilon v_{j+1}^{2}$	$\displaystyle=$	$\displaystyle L_{U,j}(\Psi)(R,v)+\varepsilon N_{U,j}(\Psi,R,v),$
	$\displaystyle\varepsilon^{2}\ddot{v}_{j+1}+2v_{j+1}(1+2\Psi_{j}+2\varepsilon R_{j})$	$\displaystyle=$	$\displaystyle\varepsilon^{2}L_{w,j}(\Psi)(R,v)+\varepsilon^{3}N_{w,j}(\Psi,R,v)+\varepsilon^{-1}\textrm{Res}_{w,j},$

where the linear terms in $(R,v)$ are given by

	$\displaystyle L_{U,j}(\Psi)(R,v)$	$\displaystyle=$	$\displaystyle\frac{1}{2}(R_{j+2}+R_{j-2})+\frac{1}{2}(v_{j+3}-v_{j-1})$
			$\displaystyle+\Psi_{j+2}(R_{j+2}+v_{j+3})+\Psi_{j-2}(R_{j-2}-v_{j-1}),$
	$\displaystyle L_{w,j}(\Psi)(R,v)$	$\displaystyle=$	$\displaystyle-(1+2\Psi_{j})v_{j+1}-\frac{1}{2}(R_{j+2}-R_{j-2})-\frac{1}{2}(v_{j+3}+v_{j-1})$
			$\displaystyle-\Psi_{j+2}(R_{j+2}+v_{j+3})+\Psi_{j-2}(R_{j-2}-v_{j-1}),$

and quadratic terms in $(R,v)$ are given by

	$\displaystyle N_{U,j}(\Psi,R,v)$	$\displaystyle=$	$\displaystyle\frac{1}{2}(R_{j+2}+v_{j+3})^{2}+\frac{1}{2}(R_{j-2}-v_{j-1})^{2},$
	$\displaystyle N_{w,j}(\Psi,R,v)$	$\displaystyle=$	$\displaystyle-2R_{j}v_{j+1}-\frac{1}{2}(R_{j+2}+v_{j+3})^{2}+\frac{1}{2}(R_{j-2}-v_{j-1})^{2}.$

Thanks again for $\ell^{2}$ being a Banach algebra with respect to pointwise multiplication, the linear and quadratic terms obey the following estimate.

The dynamics of the error functions is estimated with the help of a suitable chosen energy. We define the energy function by

(16)

$\displaystyle E(t)=\frac{1}{2}\sum_{j\in 2{\mathbb{Z}}}\dot{R}_{j}^{2}+R_{j}^{2}+\varepsilon^{2}\dot{v}_{j+1}^{2}+2v_{j+1}^{2}+2\Psi_{j}(R_{j}^{2}+2v_{j+1}^{2})+4\varepsilon R_{j}v_{j+1}^{2}.$

Computing the time derivative of $E(t)$ yields

	$\displaystyle\frac{d}{dt}E(t)$	$\displaystyle=$	$\displaystyle\langle\dot{R},\ddot{R}+R+2\Psi R+2\varepsilon v^{2}\rangle_{\ell^{2}}$
			$\displaystyle+\langle\dot{v},\varepsilon^{2}\ddot{v}+2v+4\Psi v+4\varepsilon Rv\rangle_{\ell^{2}}+\langle\dot{\Psi},R^{2}+2v^{2}\rangle_{\ell^{2}},$

where $(\Psi R)_{j}=\Psi_{j}R_{j}$ and $(\Psi v)_{j}=\Psi_{j}v_{j+1}$. By substituting the dynamical equations for $(R,v)$ into (3), we obtain

	$\displaystyle\frac{d}{dt}E(t)$	$\displaystyle=$	$\displaystyle\langle\dot{R},-\varepsilon R^{2}+\varepsilon v^{2}+L_{U}(\Psi)(R,v)+\varepsilon N_{U}(\Psi,R,v)\rangle_{\ell^{2}}$
			$\displaystyle+\langle\varepsilon\dot{v},\varepsilon L_{w}(\Psi)(R,v)+\varepsilon^{2}N_{w}(\Psi,R,v)+\varepsilon^{-2}\textrm{Res}_{w}\rangle_{\ell^{2}}$
			$\displaystyle+\langle\dot{\Psi},R^{2}+2v^{2}\rangle_{\ell^{2}}.$

The energy function controls the perturbations and their time derivative if displacements of heavy particles relatively to each other are sufficiently small, as in the condition (7) of Theorem 1. The following lemma gives the corresponding result.

The essential point in the proof of the approximation result of Theorem 1 is that the fast dynamics of $v$ can be controlled by the $\|\varepsilon\dot{v}(t)\|_{\ell^{2}}^{2}$ term in the energy bound (20) and in the energy balance equation (3). By using the Cauchy-Schwarz inequality and the estimates (12), (14), and (15), we obtain the following estimate.

We can now conclude the proof of Theorem 1. Let $S(t):=E(t)^{1/2}$. The initial bound (6) yields $S(0)\leq C_{0}$ for some $C_{0}>0$ independently of $\varepsilon\in(0,\varepsilon_{0})$. The energy balance estimate (21) can be rewritten in the form

(22)

$$\frac{d}{dt}S(t)\leq C_{1}+C_{2}S(t)+C_{3}\varepsilon S(t)^{2},\quad t\in[0,T_{0}],$$

where the constants $C_{1},C_{2},C_{3}>0$ have been redefined. Let $T_{*}$ be defined by

$$T_{*}:=\sup\{T>0:\quad S(t)\leq\varepsilon^{-1}C_{3}^{-1}C_{2},\quad t\in[0,T]\},$$

for the given constants $\varepsilon$, $C_{2}$, and $C_{3}$. Then, by Gronwall’s inequality, we obtain

$$S(t)\leq\left[S(0)+(2C_{2})^{-1}C_{1}\right]e^{-2C_{2}t}\leq\left[C_{0}+(2C_{2})^{-1}C_{1}\right]e^{-2C_{2}T_{0}},\quad t\in[0,T_{0}].$$

Since $T_{0}<T_{*}$ if $\varepsilon>0$ is appropriately small, we obtain $S(t)\leq C$ for some $C>0$ independently of $\varepsilon\in(0,\varepsilon_{0})$, so that the final bound (8) holds. The approximation result of Theorem 1 is proven.

We have proven the approximation result of Theorem 1 for the simplest nonlinear interaction potential $W^{\prime}(u)=u+u^{2}$. For a more general interaction potential $W\in C^{3}(\mathbb{R})$, Taylor expansions around $U$ yield

$$W^{\prime}(U_{j}-w_{j+1})+W^{\prime}(U_{j}+w_{j+1})=2W^{\prime}(U_{j})+\mathcal{O}(|w_{j+1}|^{2})$$

and

$$W^{\prime}(U_{j}-w_{j+1})-W^{\prime}(U_{j}+w_{j+1})=-2W^{\prime\prime}(U_{j})w_{j+1}+\mathcal{O}(|w_{j+1}|^{2}),$$

so that the system of coupled equations (9) and (10) is rewritten in the more general form:

(23)		$\displaystyle\ddot{U}_{j}+W^{\prime}(U_{j})+\mathcal{O}(|w_{j+1}|^{2})$	$\displaystyle=$	$\displaystyle g(U_{j+2},U_{j-2},w_{j+3},w_{j-1}),$
(24)		$\displaystyle\varepsilon^{2}\ddot{w}_{j+1}+(2+\varepsilon^{2})W^{\prime\prime}(U_{j})w_{j+1}+\mathcal{O}(|w_{j+1}|^{2})$	$\displaystyle=$	$\displaystyle\varepsilon^{2}h(U_{j+2},U_{j-2},w_{j+3},w_{j-1}),$

The energy function for the perturbation terms in the decomposition (13) becomes

(25)

$\displaystyle E(t)=\frac{1}{2}\sum_{j\in 2{\mathbb{Z}}}\dot{R}_{j}^{2}+W^{\prime\prime}(\Psi_{j})R_{j}^{2}+\varepsilon^{2}\dot{v}_{j+1}^{2}+2W^{\prime\prime}(\Psi_{j}+R_{j})v_{j+1}^{2}.$

It follows by repeating the previous analysis that the same approximation result stated in Theorem 1 applies to the more general interaction potential satisfying the conditions $W\in C^{3}(\mathbb{R})$ and $W^{\prime\prime}(0)>0$.