Breather solutions for inhomogeneous FPU models using Birkhoff normal forms

Highlights

• Derive inhomogeneous FPU model from elastic networks in multidimensions.

• Show examples of spatially localized normal modes in linearized FPU with spatial inhomogeneities.

• Show possibility of continuation of localized linear modes using Birkhoff normal forms.

• Compute numerically periodic orbits in quartic Birkhoff normal form, relate to spatial localization.

• Examples include localized oscillations in 3-D model with protein geometry.

Abstract

We present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of Fermi–Pasta–Ulam (FPU) type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations. The main feature of the FPU lattices we consider is that the number of interacting neighbors varies from site to site, and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem. This property is seen in 1-D models, and in a 3-D model with a geometry obtained from protein data. The spectral analysis of these examples suggests some non-resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic Birkhoff normal forms of the FPU systems. The invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way.

Introduction

We derive and study a class of nonlinear lattices of Fermi–Pasta–Ulam (FPU) type with a site dependent number of interacting neighbors. Our results indicate the possibility of spatially localized weakly nonlinear modes, in particular the persistence of localized linear modes that result from spatial inhomogeneities of the lattice.

The motivation comes from the question of energy localization in protein vibrations. The starting point of our analysis is the quartic nonlinear “elastic network” model of protein vibrations proposed by Juanico, Sanejouand, Piazza, and De Los Rios  [1]. Elastic networks are systems of point particles interacting via spring-like forces and have been used to study vibrations of proteins around equilibrium configurations that are considered known. The idea is to replace the complicated inter-particle potentials used in molecular dynamics by simpler, pairwise potentials that are discrete analogues of the potential energies of classical elasticity. The reference equilibrium configuration is obtained either as an equilibrium of a molecular dynamics model, or from crystallographic data for the positions of the atoms. Also, the pairwise elastic interaction only occurs between masses within a finite phenomenological interaction radius. Tirion  [2] showed earlier that linear elastic networks can capture some features of low frequency protein vibration modes, while Juanico et al.  [1], and Piazza and Sanejouand  [3], [4] added a quartic nonlinearity to the pairwise interactions and observed evidence for periodic solutions of high amplitude that are localized in regions with more pairwise interactions. The nonlinear model was justified by comparisons to molecular dynamics simulations  [1]. Details of periodic orbit computations are in  [3], [4].

In the present paper we look for small amplitude periodic oscillations that could be continued from spatially localized normal modes of the linearized problem. We start from two hypotheses, first that the inhomogeneous geometry of the model leads to the existence localized linear normal modes, and second that resonance arguments involving the linear frequencies and the localization properties of some normal modes can be used to construct periodic orbits in a suitable Birkhoff normal form of the system. To examine this scenario we derive an additional simplification of elastic network models by generalized quartic FPU-like systems in which every site interacts with a different number of other sites. The interactions between sites are described by a connectivity matrix that is determined by the geometry of the network, i.e. the positions of the masses in the equilibrium configuration, and the interaction radius of the elastic network. Our strategy is furthermore to first consider the question of localization in 1-D lattices, and then examine possible generalizations of some results obtained in these toy models to higher dimensions, especially lattices obtained from protein crystallographic data.

Starting with 1-D models, we examine oscillations around equilibrium configurations with variable density. Particles in higher density (or “agglomeration”) regions interact with more neighbors. In the simplest case where we have one such higher density region we see that the highest frequency linear normal modes are localized precisely in that region. Moreover the highest frequencies are separated from the other frequencies by a “gap”, see  [5]. In the presence of more agglomeration regions we see that the gap is generally filled, while the localization of linear normal modes in the agglomeration regions persists. 3-D lattices modeling protein geometries contain several regions of higher density and can exhibit similar linear localization phenomena, without frequency gaps. In this work we study a 3-D lattice representing the protein Ribozyme.

The results on the linearized FPU lattice are in Section  2.

The frequency gaps observed in 1-D models can be used to show the existence of periodic orbits in a partial quartic Birkhoff normal form of the nonlinear system, see  [5]. The periodic orbits belong to an invariant subspace of the quartic normal form that is spanned by high frequency modes that can be also spatially localized by the results of Section  2.

In the present work we generalize this argument to models without a frequency gap. To do this we examine the coefficients of quartic terms that describe the interaction between high frequency modes, and modes in a suitably defined medium frequency range. We show that the small size of these coefficients can cancel the effect of possible near-resonances and use this analysis to state a result on the existence of an invariant subspace of high frequency modes, Proposition 3, Section  3.

The calculation of periodic orbits in the invariant subspace is simplified by the fact that the restriction of the normal form to the invariant subspace has a global phase symmetry. This extra symmetry is a consequence of the small frequency width of the corresponding modes, and allows us to prove the existence of certain types of periodic orbits of “breather” type, see e.g.  [6], [7], using an elementary variational argument that also leads to a simple method to compute the periodic orbits numerically. Results for 1-D models and the Ribozyme are in Section  4.

In summary, the above theory is a step towards explaining spatial localization in the regions suggested by  [1], [3], [4], but for small amplitudes. Possible extensions to high amplitudes are discussed in Section  5. Also, the paper combines theoretical statements with assumptions supported by numerics, and leaves several questions we hope to address in further work. A fully theoretical study may be possible for some 1-D models.

We note that there is an extensive literature on spectral localization at low frequencies for the homogeneous FPU lattice, see  [8], [9] for periodic orbits (also known as q-breathers),  [10], [11] for tori. 2- and 3-D extension have been examined in  [12], also for a homogeneous lattice. Results on the integrability of Birkhoff normal forms for the homogeneous 1-D FPU system in  [13], [14], [15] give further information on the stability of small amplitude spectrally localized solutions. The flow of energy to higher modes can be also effectively controlled by only a few adiabatic invariants found by normal forms, see  [16], [13] for a related idea. Here we are concerned with the continuation of high frequency normal modes with spatial localization. In this case the mode coupling coefficients can have properties not seen in homogeneous problems (see  [12] for high frequency modes of the homogeneous FPU). Numerical simulations supporting the scenario of energy localization at high frequencies will be presented elsewhere and are not directly explained by the present study. The FPU approximation and elastic network models are briefly discussed in Sections 2, and 5. We believe that the question of more general interactions should be studied further. Also, there are classical existence results on periodic orbits near elliptic equilibria, e.g. the Weinstein–Moser theorem  [17], [18], see also  [19], the Lyapunov center theorem, see e.g.  [20], and an extensive literature on periodic orbit calculations. Some relevant points are discussed in Section  5.

The paper is organized as follows. In Section  2 we present the quartic elastic model and describe the steps leading to the FPU-type model (Section  2.1). In Section  2.2 we decompose the linearized FPU system into noninteracting normal modes. Numerical examples of normal modes are shown in Section  2.3.

Section  3.1 reviews Birkhoff normal forms, explaining our notation. In Section  3.2 we discuss normal forms and invariant subspaces for lattices with frequency gaps, and in Section  3.3 we extend these arguments to cases without gap. We apply these ideas to 1-D lattices and to the Ribozyme 3-D lattice. In Section  4 we calculate some periodic orbits of breather type in the invariant subspace and check their linear stability. Examples include the Ribozyme. In Section  5 we discuss our results.