User:Eugene M. Izhikevich/Proposed/Breathers

• Sergej Flach, Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany

• Andrey Gorbach, Department of Physics, University of Bath, UK

Dr. Sergej Flach accepted the invitation on 2 November 2010

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Discrete Breathers, also known as Intrinsic Localized Modes, are localized excitations in spatially discrete systems with perfect translational invariance. They have been theoretical constructs for more than a decade. Only recently have they been observed in physical systems as distinct as charge-transfer solids, Josephson junctions, photonic structures, micromechanical oscillator devices, ultracold atoms in optical lattices, and antiferromagnetic layered structures. They provide deep insights into the complex dynamics of nonlinear discrete systems and may lead to novel devices for energy trapping, storage and transfer, as well as for information processing.

Introductory remarks

In solid state physics, the phenomenon of localization is usually perceived as arising from extrinsic disorder that breaks the discrete translational invariance of the perfect crystal lattice. Familiar examples include the localized vibrational phonon modes around impurities in crystals and Anderson localization of electrons in disordered media. The typical perception is that in perfect lattices, free of extrinsic defects, phonons and electrons exist only in extended, plane wave states. Using the Bloch or Floquet theorems, similar arguments would apply to any periodic structure, such as a photonic crystal or a periodic array of optical waveguides. These firmly entrenched perceptions were severely jolted in the late 1980s by the discovery of discrete breathers or intrinsic localized modes as typical excitations in perfect but strongly nonlinear, spatially extended discrete periodic systems.

A good working definition of discrete breathers is that they are spatially localized, time-periodic excitations in spatially extended, perfectly periodic, discrete systems.

Since their discovery, discrete breathers have become the subject of intense theoretical and experimental interest. This flurry of activity and the limitations of space imply that our discussion is necessarily incomplete, and readers are urged to consult some of the several pioneering papers, recent reviews, and websites for more details.

Despite the surprise that accompanied their discovery, there had been clear historical precedents for the phenomenon of intrinsic localization. A long history of developments established that nonlinearity alone could produce localized states but also that spatially localized, time-periodic solutions - breathers - were very rare in the continuum limit (where the discreteness of the underlying system may be safely ignored). The best approach for establishing the existence of discrete breathers in spatial lattices is to start from a system of uncoupled nonlinear oscillators - the anti-continuum limit - and treat the coupling as a perturbation. Let us explore in more detail how these two insights led naturally to a systematic understanding of and approach to discrete breathers.

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Intuition and theory

One anharmonic oscillator

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Spatial discreteness and nonlinearity

Consider a one-dimensional chain of interacting (scalar) oscillators or atoms with the Hamiltonian \(\tag{1} H = \sum_n \left[ \frac{1}{2}p_n^2 + V(x_n) + W(x_n - x_{n-1}) \right]\;. \)

The integer \(n\) marks the lattice site number of a possibly infinite chain, and\(x_n\) and \(p_n\) are the canonically conjugated coordinate and momentum of a degree of freedom associated with site number \(n\) The on-site potential \(V\) and the interaction potential \(W\) satisfy \(V'(0)=W'(0)=0\) \(V''(0),W''(0) \geq 0\ .\) This choice ensures that the classical ground state \(x_n=p_n=0\) is a minimum of the energy\(H\ .\) The equations of motion read \(\tag{2} \dot{x}_n=p_n\;,\;\dot{p}_n=-V'(x_n) -W'(x_n - x_{n-1}) + W'(x_{n+1}-x_n)\;. \)

Let us linearize the equations of motion around the classical ground state. We obtain a set of linear coupled differential equations with solutions being small amplitude plane waves\[\tag{3} x_n(t) \sim {\rm e}^{{\rm i}(\omega_q t - qn)} \;,\; \omega_q^2 = V''(0) + 4W''(0)\sin^2 \left(\frac{q}{2}\right)\;. \]

These waves are characterized by a wave number \(q\) and a corresponding frequency \(\omega_q\ .\) All allowed plane wave frequencies fill a part of the real axis which is coined linear spectrum. Due to the underlying lattice the frequency \(\omega_q\) depends periodically on \(q\) and its absolute value has always a finite upper bound. The maximum (Debeye) frequency of small amplitude waves \(\omega_{\pi}=\sqrt{V''(0) + 4W''(0)}\ .\) The dispersion relation \(\omega_q\) is shown in . Depending on the choice of the potential \(V(x)\) it can be either acoustic- or optic-like, \(V''(0)=0\) and \(V''(0)\ne 0\) respectively. In the first case the linear spectrum covers the interval \(-\omega_{\pi} \leq \omega_q \leq \omega_{\pi}\) which includes \(\omega_{q=0}=0\ .\) In the latter case an additional (finite) gap opens for \(|\omega_q|\) below the value \(\omega_{0}=\sqrt{V''(0)}\ .\) Two further characteristics of the linear spectrum are the group velocity \(v_g\) and the phase velocity \(v_{ph}\ .\) The group velocity \(v_g(q)=d\omega_q / dq\) is a periodic function of \(q\) and describes the propagation speed of a wavepacket centered at\(q\ .\) At the edge of the linear spectrum \(v_g=0\ .\) Otherwise its absolute value has a finite upper bound. The phase velocity \(v_{ph}= \omega_q / q\) is a nonperiodic oscillating function of \(q\ .\) It covers the whole real axis for an optic-like linear spectrum since \(\omega_{q=0} \neq 0\ .\) Its absolute value has a finite upper bound \(|v_{ph}| \leq v_g(q=0)\) for acoustic-like linear spectra.

For large amplitude excitations the linearization of the equations of motion is not correct anymore. Similar to the case of a single anharmonic oscillator, the frequency of possible time-periodic excitations will depend on the amplitude of the excitation, and thus may be located outside the linear spectrum. Let us assume that a time-periodic and spatially localized state, i.e. a discrete breather, \(\hat{x}_n(t+T_b)=\hat{x}_n(t)\) exists as an exact solution of Eqs. (2) with the period \(T_b=2\pi/\Omega_b\ .\) Due to its time periodicity, we can expand \(\hat{x}_n(t)\) into a Fourier series \(\tag{4} \hat{x}_n(t) = \sum_k A_{kn}{\rm e}^{ik\Omega_b t}\;. \)

The Fourier coefficients are by assumption also localized in space \(\tag{5} A_{k,|n| \to \infty} \to 0 \;. \)

Inserting this ansatz into the equations of motion (2) and linearizing the resulting algebraic equations for Fourier coefficients in the spatial breather tails (where the amplitudes are by assumption small) we arrive at the following linear algebraic equations\[\tag{6} k^2 \Omega_b^2 A_{kn} = V''(0) A_{kn} + W''(0) (2A_{kn} - A_{k,n-1} - A_{k,n+1}) \;. \]

If \(k\Omega_b = \omega_q\) the solution to (6) is \(A_{k,n} = c_1 {\rm e}^{iqn} + c_2 {\rm e}^{-iqn}\ .\) Any nonzero (whatever small) amplitude \(A_{k,n}\) will thus oscillate without further spatial decay, contradicting the initial assumption. If however \(\tag{7} k \Omega_b \neq \omega_q \)

for any integer \(k\) and any \(q\) then the general solution to (6) is given by \(A_{k,n} = c_1 \kappa^n + c_2 \kappa^{-n}\) where \(\kappa\) is a real number depending on \(\omega_q\),\(\Omega_b\) and \(k\ .\) It always admits an (actually exponential) spatial decay by choosing either \(c_1\) or \(c_2\)to be nonzero. In order to fulfill (ref>1.2-7</ref>) for at least one real value of \(\Omega_b\) and any integer \(k\) we have to request \(|\omega_q|\) to be bounded from above. That is precisely the reason why the spatial lattice is needed. In contrast most spatially continuous field equations will have linear spectra which are unbounded. That makes resonances of higher order harmonics of a localized excitation with the linear spectrum unavoidable. The nonresonance condition (7) is thus an (almost) necessary condition for obtaining a time-periodic localized state on a Hamiltonian lattice.

The performed analysis can be extended to more general classes of discrete lattices, including e.g. long-range interactions between sites, more degrees of freedom per each site, higher-dimensional lattices etc. The resulting non-resonance condition (7) keeps its generality, illustrating the key role of discreteness and nonlinearity for the existence of discrete breathers.

Why only time-periodic orbits?

An analogous approach yields, that for a assumed quasi-periodic breather with \(N\) incommensurate frequencies \(\left\{\Omega_1,\Omega_2,...\Omega_N\right\}\) the non-resonance condition (7) transforms into \(\tag{8} \left\{k_1 \Omega_1+k_2 \Omega_2+ ... + k_N \Omega_N\right\}^2 \neq \omega^2_q \; \)

with \(k_i\) being arbitrary integer numbers. In other words, neither any of the principal frequencies \(\left\{\Omega_1,\Omega_2,...\Omega_N\right\}\) nor any linear combination of their multiples should resonate with the linear spectrum. However any incommensurate pair of frequencies \(\Omega_1\) and \(\Omega_2\) with irrational ratio \(\Omega_1/\Omega_2\) will generate an infinite number of pairs \(k_1,k_2\) which violate the non-resonance condition (8). Therefore, in general quasiperiodic breathers are not expected to exist as exact spatially localized solutions.

What about moving breathers? A rather general definition of a moving breather assumes a localized object which translates \(n\) sites in a certain direction after \(m\) periods of internal oscillations with the ratio \(n/m\)being in general irrational. A detailed analysis of possible resonances leads to the conclusion, that one has to avoid resonances of the breather velocity \(V\) with phase velocities \(v_{ph}\) of small amplitude plane waves (of sometimes modified linear spectra as compared to the original underlying one). The essence is that these resonances can not be avoided, so that moving breathers are not expected to be generic exact solutions for a general nonlinear lattice.

Examples of discrete breather solutions

Let us show discrete breather solutions for various lattices. We start with a chain (1) with the functions \(\tag{9} V(x)=x^2+x^3+\frac{1}{4}x^4\;,\;W(x)=0.1 x^2\;. \)

The spectrum \(\omega_q\) is optic-like and shown in . Discrete breather solutions can have frequencies \(\Omega_b\) which are located both below and above the linear spectrum. The time-reversal symmetry of (2) allows to search for DB displacements \(x_n(t=0)\) when all velocities \(\dot{x}_n(t=0)=0\ .\) These initial displacements are computed with high accuracy and plotted in the insets in . We show solutions to two DB frequencies located above and below \(\omega_q\) - their actual values are marked with the green arrows. To each DB frequency we show two different spatial DB patterns - among an infinite number of other possibilities, as we will see below. The high-frequency DBs (\(\Omega_b \approx 1.66\)) occur for large-amplitude, high-energy motion with adjacent particles moving out of phase. Low-frequency DBs (\(\Omega_b \approx 1.26\)) occur for small-amplitude motion with adjacent particles moving in phase.

In we show two breather solutions for a Fermi-Pasta-Ulam chain of particles coupled via anharmonic springs \(V(x)=0, W(x)=\frac{1}{2}x^2+\frac{1}{4}x^4\) which has an acoustic-type spectrum. The breather frequency is in both cases \(\Omega_b=4.5\ .\) Again the displacements \(x_n\) are shown for an initial time when all velocities vanish. In the inset we plot the strain \(u_n=x_n-x_{n-1}\) on a log-normal scale. The breather solutions are exponentially localized in space.

Finally we show breather solutions for a two-dimensional square lattice of anharmonic oscillators with nearest neighbour coupling. The equations of motion read \(\tag{10} \ddot x_{i,j}=k(x_{i+1,j}+x_{i-1,j}-2x_{i,j}) +k(x_{i,j+1}+x_{i,j-1}-2x_{i,j}) -x_{i,j}-x_{i,j}^3 \)

with oscillator potentials \(V(x)=\frac{1}{2}x^2+ \frac{1}{4}x^4\ .\) In we plot the oscillator displacements with all velocities equal to zero for three different DB frequencies and \(k=0.05\ .\) For all cases adjacent oscillators move out of phase.

Discrete breather solutions can be typically localized on a few lattice sites, regardless of the lattice dimension. Thus little overall coherence is needed to excite a state nearby - just a few sites have to oscillate coherently, the rest of the lattice does not participate strongly in the excitation.

Existence proofs

Anti-continuum limit: From the present perspective the most important result in this field is due to MacKay and Aubry. They consider d-dimensional lattice models of coupled anharmonic oscillators (cf. e.g. Eq.(10) for \(d=2\)) in the limit of very weak interaction (anticontinuous limit). For zero interaction the oscillators decouple, and it is trivial to construct arbitrary spatially localized excitations. MacKay and Aubry choose those trajectories which are periodic in time, and time-reversible. These trajectories can be encoded by a sequence of time-periodic and time-reversible functions, where each of them is representing the time-dependence of the coordinate of an oscillator. Note that the period is one and the same for all functions. The sequence can be formally represented as a \(N\)-dimensional vector, where \(N\) is the number of oscillators, and each component is a time-periodic and time-reversible function. The vector can be embedded in a corresponding Banach space.

The original equations of motion are used to define a map of a vector from that Banach space onto another vector. Solutions of the equations of motion are those vectors which are mapped into the zero (origin) of the space. For zero interaction the simplest example is when one oscillator is excited and all others are at rest.

One of the most elegant parts of the existence proof of MacKay and Aubry is the use of the Implicit Function Theorem in order to show that when the interaction is small but nonzero, there is a new vector in the neighbourhood of the old one, which is still mapped into the origin, and thus there is a solution of the equations of motion. And moreover they use the norm properties of vectors from Banach space to show that the still existing but deformed vector solution corresponds to an exponentially localized excitation on the original lattice. Remarkably the necessary condition for the proof to work is the nonresonance condition (7).

The power of that existence proof is that it is essentially insensitive to the lattice dimension, the type of interaction on the lattice, and the number of originally excited sites. Later, the method has been modified to prove persistence of non-time-reversible periodic orbits as well.

Anharmonic interactions: Switching off the oscillator potentials, and leaving all the nonlinearity in the interactions makes things different. The above discussed proof does not apply. The first proof of existence of DBs in such one-dimensional systems used the special case of homogeneous interaction potentials, which allow for a separation of time and space variables . Livi, Spicci and MacKay considered more general potentials, but used the limit of strongly alternating particle masses instead . In the limit of infinite (or zero) mass ratio the lattice dynamics is reduced to a special anticontinuous case of light masses oscillating between immobile heavy masses. With the help of the Implicit Function Theorem localized vibrations were continued into the regime of finite mass ratio.

Another possibility is to consider breathers having frequencies close to the linear spectrum. Aubry et al. performed a variational approach to rigorously prove the existence of DBs for essentially any lattices with pure convex interaction potentials . James performed a center manifold reduction and proved the existence of weakly localized DBs in one-dimensional FPU chains (see also Ref. for an extensive review of that technique).

Applicability: Most of the above proofs are implicit, i.e. the existence of DBs is proven without explicit construction of the solutions. However, the case of weak interactions, and the limit of frequencies close to the linear spectrum, defines the precise form of the solutions in that very limit. Therefore, numerical continuation tools harvest on that information - they use the exact solution in the limit, the knowledge that it can be continued away from the considered limit, and therefore successfully generate solutions further and further away.

Basic properties of discrete breathers

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Applications

There is a fastly growing amount of experimental and related theoretical work on discrete breathers in many different branches in physics, like superconducting materials, Bose-Einstein condensates, antiferromagnetic structures, crystals and molecules, micromechanical systems, and others. Below we briefly discuss some of the most common setups.

Coupled optical waveguides

Coupled optical dielectric waveguides form periodical structures in one and two dimensions [..]. Each individual waveguide is a narrow channel (typical widths are of the order of several micrometres) surrounded by a dielectric with lower index of refraction. Light guidance inside the channel occurs due to the total internal reflection at interfaces and is mathematically equivalent to a quantum particle trapping in the potential well formed by the contrast in the refractive index \(U_{eff}=-n^2(\vec{r}_\bot)\) (regions with higher refractive index correspond to lower effective energies). Placing several waveguides close to each other, light can couple from one waveguide to the neighboring ones due to the overlap of evanescent tails of individual waveguide modes - similar to the quantum particle tunneling in a multi-well potential. Nonlinearity in the system is due to the dependence of the refractive index of the medium on local light intensity (Kerr effect). Under certain approximations, light propagation in a coupled waveguide structure can be described with complex-valued amplitudes of individual waveguide modes \(E_\mathbf{k}(z)\ ,\) slowly varying (as compared to the carrier wavelength) with propagation distance \(z\ .\) Here \(\mathbf{k}\) is the waveguide index (set of indices) in the corresponding one- or two-dimensional lattice. The system is eventually reduced to the set of coupled Discrete Nonlinear Schrödinger (DNLS) equations for the amplitudes \(E_\mathbf{k}(z)\ :\) \[\tag{11} i\frac{dE_\mathbf{k}}{dz}+\sum_{\mathbf{m}\in \mathbf{N}_k}C_{\mathbf{k},\mathbf{m}}E_\mathbf{m}+F(|E_\mathbf{k}|^2)E_\mathbf{k}=0 \ .\]

Here \(\mathbf{N}_k\) is the set of indices corresponding to the nearest neighbors of site \(\mathbf{k}\) (e.g. in 1D lattice \(\mathbf{N}_k=\{k+1,k-1\}\ ,\) in 2D square lattice each site is labeled by two indices \(\mathbf{k}=[k_1,k_2]\ ,\) and \(\mathbf{N}_k=\{[k_1+1,k_2], [k_1-1,k_2], [k_1,k_2+1], [k_1,k_2-1]\}\)). For a perfectly periodic lattice of identical waveguides coupling coefficients \(C_{\mathbf{k},\mathbf{m}}\) are all equal and indices can be omitted. At low light intensities Kerr nonlinearity can be approximated as \(F(|E_\mathbf{k}|^2)=\gamma|E_\mathbf{k}|^2\ ,\) higher order expansion terms may become important to describe e.g. saturation of nonlinearity.

In the above system, the evolution coordinate is propagation distance \(z\) along waveguides, while electromagnetic field is stationary in time. In a typical experiment a single waveguide is excited by a focused laser beam at the input facet (\(z=0\)), and light distribution across the lattice is monitored at the output facet (\(z=L>L_C\)) for gradually increasing input powers. Waveguide length \(L\) (typically ranging from few mm to few cm) is assumed to be larger than the coupling length \(L_C=\pi/C\) over which light tunnels from the input waveguide to its neighbors when nonlinearity is disregarded (low powers). For high input powers one observes localization of light in the input waveguide, thus confirming excitation of a discrete breather in the system of Eqs. (11).

We note that the specific type of nonlinearity in Eqs. (11) allows single-harmonic \(z\)-dependence \(E_{\mathbf{k}}(z)=f_{\mathbf{k}}e^{iqz}\ ,\) where \(f_{\mathbf{k}}\) solve the corresponding system of algebraic equations. This is why discrete breathers in optics are often coined 'discrete solitons'.

Different one- and two-dimensional optical waveguide lattices, both static and reconfigurable, are relatively easily manufactured or imprinted [..], and represent a perfect setting for direct observation of different types of discrete breathers and associated dynamics.

Antiferromagnetic layered structures

Spin waves in magnetically ordered media are traditionally considered as a good experimental setup for studies of nonlinear wave dynamics [..]. Both interactions between spins and spin anisotropy are intrinsically nonlinear, and dissipation of spin waves is typically quite weak, as compared e.g. to phonon dissipation in crystals.

In a layered antiferromagnetic crystal, below the Neel temperature, spin 1/2 ions are oriented along the easy-axis in alternating sheets.

Subsection b

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Subsection c

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Section d

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Citing references

Groups of authors larger than 2 can be cited with "et al.".

• As proven in (Albero A, 1999).
• As Albero (2009) said.
• As proven in (Albero and Bocca, 2001)
• As proven by Albero and Bocca (2001)
• As proven by Albero et al. (2003)
• As proven by Albero, Bocca and Cuoco (2003)
• As proven by Albero et al. (2007a), confirmed by Albero et al. (2007b) and discarded by Albero et al. (2007c)
• As proven in (Albero A, 1999).
• As proven by Albero and Bocca (2001).

References

• Albero, Antony (1999). Pizza Margherita. Journal of pizza eaters 19(3): 13. arXiv:0808.000
• Albero(2001). Pizza Capricciosa. Journal of pizza eaters 27: 121-127. arXiv:0808.000
• Albero, Antonio; Bocca, Bill and Cuoco, C T (2003). Pizza Quattro Stagioni. Journal of pizza eaters 34(4): 12.
• Albero, Antonio; Bocca, Bill; Cuoco, C T and Dude, David B (2007a). Pizza Napoletana. Journal of pizza eaters 37: 121-127.
• Albero, Antonio; Bocca, Bill; Cuoco, C T; Dude, David and Elica, E Q (2007b). Pizza Marinara. Journal of pizza eaters 43(4): 1-13.
• Albero, Antonio et al. (2008). Pizza Piccante. Journal of pizza eaters 45(5): 1-13.
• Alto, Antony (1999). La Pizza! Mangiare bene, Volume 3. Albero and Bacca editors. Food Publishers, Genoa.
• Alto, Antony and Bocca, Bill (2000). La Pasta! Mangiare bene. Albero editor. Food Publishers, Genoa. Chapter 1.
• Alto, Antony; Bocca, Bill and Cuoco, C T (2002). Pizza: prepare it yourself. Food Publishers, Genoa. Page 22. ISBN 1-234-99929-0.
• Alto, Antony; Bocca, Bill; Cuoco, C T and Dude, David B (2005a). Italian Pizza. Food Publishers, Genoa.
• Alto, Antony; Bocca, Bill; Cuoco, C T; Dude, David B and Elica, E Q (2005b). Napolitan Pizza. Food Publishers, Genoa.
• Alto, Antony et al. (2005c). American Pizza. Food Publishers, Genoa.

Further reading

• Magro, C T (2008). Pizza: a danger for health? Food Publishers, Paris. page 22. ISBN 1-234-90929-0. This reference is unreliable in conclusions, but quite accurate in its introduction.
• Izhikevich, E M (2007). Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting The MIT Press, Cambridge, MA. ISBN 0262090430. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience.

External links

Eugene M. Izhikevich website

See also

Brain, Neuron, Scholarpedia:Instructions for authors