Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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approximated by the expression in action-angle

variables

H ¼ H

þ H

 !



ð!

½4

Here, J

= !

is the action variable. In practice,

this amounts to approximate the original Hamilto-

nian by the sum of the harmonic and nonlinear self-

energy of the initi ally excited mode. In this frame-

work, H

and H

are the unperturbed (integrable)

Hamiltonian and the perturbation, respectively.

Indeed, if the energy is initially attributed to mode

k, the following relations hold: !

 H

 E.By

the approximated Hamiltonian [4], one can com-

pute the nonlinear correction to the linear frequency

, giving the renormalized frequency !

¼ !



¼ !

þ 

½5

For N  k one has





H

½6

The distance between two primary resonances, in

the harmonic limit, is given by the expression

!

¼ !

kþ1

 !

 N

1

½7

Consistently with [6], the last approximation is valid

only for small wave number (k  N), that is, long-

wavelength modes.

The ‘‘resonance overlap’’ criterion amounts to

compare this distance with the frequency shift. In

formulas:



 !

½8

This equation allows to obtain also an estim ate of

the ‘‘critical’’ energy density, 

, above which size-

able chaotic regions develop and a fast diffusion

takes place in phase space:







k

½9

with k = O(1)  N. Below 

, primary resonances

are weakly coupled and determine a slow-relaxation

process to energy equipartition. Above 

, due to

‘‘primary resonance’’ overlap, fast relaxation to

equipartition sets in (Izrailev and Chirikov 1966).

This prediction was verified numerically later by

Chirikov et al. (1973). The presence of a critical

energy density can be tested by meas uring the

evolution of the finite time-averaged quantity



(t) = t

1

()d, where E

= (P

þ !

)=2is

the harmonic energy of the kth mode. For energy

densities much smaller than 



(t) exhibits an

extremely slow relaxation towards the equipartition

condition,



= constant. Conversely, for >

such

a condition is rapidly approached on a relatively

short timescale. The slow relaxation below 

can be

traced back to the overlap of higher-order reso-

nances: its typical timescale has been found to be

inversely proportional to a power of the energy

density (Shepelyansky 1997).

Energy-Equipartition Thresholds

The first paper reporting evidence of the existence of

an energy threshold in chains of coupled anharmo-

nic oscillators had already been published in 1970

by Bocchieri et al. (1970). This pioneering numerical

experiment concerned a chain of oscillators coupled

through a Lennard-Jones interatomic potential. The

Italian group observed an energy threshold, separat-

ing a high-energy thermalized regime from a regular

dynamics regime at low energies (like the one

observed by Fermi, Pasta, and Ulam). The main

point raised by this experiment concerns the

consequences on ergodic theory: the ordered motion

observed in the low-energy regime seems to violate

ergodicity, although the model is known to be

chaotic at any energy.

This is quite a delicate and widely debated issue

for its statistical implications. Actually, as we have

mentioned in the previous section, also Fermi, Pasta,

and Ulam expected that a nonlinear dynamical

system, made of a large number of degrees of

freedom, should naturally evolve towards equili-

brium. Further confirmations to the seminal paper

by Bocchieri and co-workers came from more

refined numerical experiments, showing that, for

sufficiently high energies, regular behaviors disap-

pear, while equipartition among the Fourier modes

sets in rapidly. Later on, the presence of the energy

threshold was characterized by introducing an

appropriate entropy, S = 

ln p

with p

(t) =Ei, which counts the number of effective

Fourier modes involved in the dynamics: at equi-

partition, this entropy is maximal (Livi et al. 1985).

Nowadays, we know that the approach to

equipartition below and above the energy threshold

is a matter of timescales, which turn out to be very

different in the two regimes. For instance, the

analytic estimate of the maximum Lyapunov expo-

nent  of the FPU -model (Casetti et al. 1995) has

definitely pointed out that there is a threshold value

of the energy density, 

, at which its dependence on

 changes drastically:

ðÞ



1=4

if   

;



if   



½10

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations 547

This implies that the typical relaxation time, that is



1

, may become exceedingly large for very small

values of  below 

. It is worth stressing that this

result holds in the thermodynamic limit, thus indicat-

ing that the presence of 

is statistically relevant.

A more controversial scenario emerges from the

studies of the relaxation dynamics for specific

classes of initial conditions. When a few long-

wavelength modes are initially excited, regular

motion may persist over times much longer than



1

(De Luca et al. 1995). The excitation of small-

wavelength modes yields an even more complex

scenario: solitary wave dynamics is observed, fol-

lowed by slow relaxation to equipartition. It is also

worth mentioning that some regular features of the

dynamics persist even at high energies. As we shall

discu ss in the section ‘‘Heat trans port,’’ such

regularities still play a crucial role in determining

energy transport mechanisms, although they do not

affect significantly the equilibrium statistical proper-

ties of the FPU model at high energies.

The Generalized Fluctuation–Dissipation

Theorem

Another fundamental problem of nonequilibrium

statistical mechanics concerns the possibility of

establishing a fluctuat ion–dissipation theorem, gen-

eralizing the relation valid for equilibrium condi-

tions. In fact, on this basis one might develop a

large-deviation formalism, aiming at the identifica-

tion of an explicit nonequilibrium statistical mea-

sure, analogous to the equilibrium Boltzmann–Gibbs

measure. Recently, some relevant progresses in this

direction have been made.

A crucial numerical experiment, which attract ed

the attention on the problem of formulating a

generalized fluctuation–dissipation relation for sta-

tionary flows, was performed at the beginning of the

1990s (Evans et al. 1993). Stationary conditions for

momentum transport were obtained in the shear

flow of a fluid contained between moving walls. The

reversibility of the microscopic dynamics yields the

heuristic fluctuation relation:

Prð



¼ AÞ

Prð



¼AÞ

¼A ½11

where Pr(



= A) is the probability that the average

entropy production rate,



, along a trajectory

segment of duration t, takes the value A. For

sufficiently large values of t, this relation was

confirmed by numerical analysis.

Gallavotti and Cohen (1995a,b) proved a theo-

rem meant to put on a rigorous mathematical

basis eqn [11], that is, the proposed extension to

nonequilibrium steady states of the equilibrium

fluctuation–dissipation theorem. This theorem

concerns the phase-space contraction rate of the

dynamics, which equals the entropy production

rate in the case of particle systems, whose internal

energy is a c onstant of the motion. The proof of

the theorem is based on restrictive hypotheses,

which include the existence of an average non-

vanishing phase-space contraction rate, the time-

reversal invariance of the dynamics and a s trong

form of chaos (the dynamics is assumed to be of

the A nosov type, that is, smooth and uniformly

hyperbolic). Nonetheless, the prediction of the

theorem, that is,



ðpÞ



ðpÞ

¼ Dhip ½12

is expected to hold much more generally. Here 

ðpÞ

is the probability that a fluctuation variable takes

the value p. The theorem proved by Gallavotti and

Cohen states that 

ðpÞ has to satisfy the large

deviation relation [12], where  is the average

phase-space contraction rate over a trajectory seg-

ment of duration t and D is a suitable constant. It

must be pointed out that the rigorous derivation of

this relation provided strong motivations for inves-

tigating its validity and generality in many other

contexts. The first numerical experiment, where

almost all the constituent hypotheses of the Gallavotti–

Cohen theorem were satisfied, was performed by

Bonetto et al. (1997). They s tudied a Lorentz gas

(massive pointlike noninteracting particles bouncing

elastically on circul ar scatter ers displac ed on a

regular lattice without free horizon) of charged

particles moving in an uniform external electr ic

field. Numerical simulations were found to be in

very good agreement with [11] and [12] (w hich, in

this case, refer to the same quantity). One further test

of the fluctuation–dissipation relation was later

performed for a different setup (Lepri et al. 1998).

The FPU -model is put in contact at its boundaries

with thermal heat baths of different temperatures T

and T



> T



). Numerical simulations have been

performed for sufficiently large applied thermal

gradients, which guarantee sizeable effects of fluc-

tuations, suitable for verifying a relation like [11].It

is worth noticing that many of the constituent

hypotheses of the Gallavotti–Cohen theorem are

not valid for this setup, but eqn [12] is still expected

to hold, although in this case it does not refer to the

entropy production rate. Nonetheless, the extension

[11] of the fluctuation–dissipation theorem can be

tested, thanks to the following useful relation,

548 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

between the heat flux j and the entropy production

rates, 



, at the chain boundaries:

h

iþh



i¼j





½13

This can be interpreted as a balance relation for the

global entropy production. In fact, according to the

principles of irreversible thermodynamics, the local

rate of entropy production  in the bulk is given by

ðxÞ¼j

TðxÞ



½14

By integrating this equation, one straightforwardly

obtains the previous one, which then applies to the

entropy production from the heat baths. Careful

numerical simulations show that stationary condi-

tions are found to hold over a wide range of

temperatures and gradi ents. Equation [13] indicates

that the heat flux is equivalent to the entropy

production rate, apart from a multiplicative con-

stant which depends on the amplitude of the applied

field.

Let us define the finite-time average of the global

heat flux

i¼1

dj

ðÞ½15

The normalization of this quantity can be obtained

by computing the asymptotic average value

¼ lim

t!1

½16

The quantity of statistical interest is the normalized

finite-time average global heat flux

z ¼



½17

Accordingly, the fluctuation–dissipation relation in

this case takes the form:



ðzÞ



ðzÞ

¼ zj





½18

The conje cture that such a relation might be valid in

this case has been confirmed by numerical analysis.

It is worth stressing that, in this out-of-equilibrium

setup, the probability distribution, P



(z), is not

Gaussian and exhibits a peculiar asymmetric shape.

Nonetheless, for increasing values of , the asym-

metry progressively reduces, while P



(z) approaches

a Gaussian shape. This observation indicates that, in

this case, large fluctuations deviate from the typical

statistics of independent events.

It should be mentioned that generalized fluctuation–

dissipation relations, like those discussed in this

section, have been successfully checked in many other

situations, where the hypotheses of the Gallavotti–

Cohen theorem did not apply. The ‘‘robustness’’ of

relations such as [11] and [12] indicates that a more

general theory may be possible.

Heat Transport

The validity of Debye’s conjectur e about the

necessity of nonlinear forces for obtaining a finite

heat conductivity in crystals still remained an open

problem after the unsuccessful FPU numerical

experiment. The setup, described in the previous

section for testing the generalized fluctuation–

dissipation relation in the FPU chain, can be used

also for tackling the verification of this conjecture.

Actually, the thermal conductivity, , of a chain of

oscillators can be measured from the Fourier’s law

¼rTð xÞ½19

where J

is the heat current and rT(x) is the

temperature gradient.

This problem was solved analytically for a chain

of N harmonic oscillators (Rieder et al. 1967). The

bulk of the chain is found to reach thermal

equilibrium conditions at the average temperature

T = (T

þ T



)=2, corresponding t o a constant tem-

perature profile. Only at the chain boundaries the

harmonic chain exhibits a steep temperature gra-

dient. This implies that the heat current is propor-

tional to the temperature difference, rather than to

the temperature gradient, thus violating Fourier’s

law. Accordingly, a harmonic chain, made of N

oscillators, in contact with two heat reservoirs at

different temperatures, exhibits anomalous trans-

port properties and the effective thermal conduc-

tivity is found to diverge in the infinite-chain limit

as   N. This peculiar behavior is a consequence

of the integrability of the harmonic chain

dynamics. Actually, the Fourier modes propagate

with finite velocity through the harmonic chain, so

that any energy injected from the hot reservoir

flows ballistically to the cold one, rather than

diffusing, as required for the validity of [19].Itis

worth stressing that any i ntegrable system should

exhibit a similar scenario. This is the case of the

equal-mass hard sphere gas in one dimension a nd

of the Toda chain, where the harmonic potential

=2)(q

iþ1

 q

)

is replaced by the nonlinear

expression

a exp½bðq

iþ1

 q

Þ

In the former case, integrability and ballistic

propagation are straightforward consequences of

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations 549

the conservation laws, inherent elastic collisions

between hard spheres. In the latter model, the

normal nonlinear modes, called ‘‘Toda solitons,’’

are responsible for such anomal ous behavior.

Debye’s conjecture should be modified accord-

ingly: nonintegrability of the equations of motion

has to be invoked as a necessary property for

explaining heat transport in real solids. Let us

observe that the FPU model is known not to be

integrable and it is expected to be a good candidate

for confirming Debye’s conjecture, at least in its

fully chaotic regime. Careful and extended numer-

ical simulations have shown that the FPU chain

maintains anomalous properties (Lepri et al. 1997).

In particular, the thermal conductivity, , is found

to diverge in the infinite chain limit as

  N



½20

with   2=5. This value agrees with independent

analytic estimates (e.g., see Lepri et al. (2003)),

although renormalizat ion arguments indicate that

one should rather find  = 1=3(Narayan and

Ramaswamy 2002). This discrepancy could be due

to the peculiar features associated with the presence

of a quartic nonlinearity in the FPU problem and

also to the fact that in the FPU chain heat can be

transported only through longitudinal oscillations.

Anyway, this is still an open problem, which

requires further theoreti cal advances to be solved.

In a more general perspective, the main outcome

of these numerical studies indicates that a power-

law divergence like [20] is found in all one-

dimensional nonintegrable models. This general

feature must be attributed to the combined effect

of low-space dimensionality, with energy and

momentum conservation. In such a situation,

fluctuations are strongly constrained, so that the

evolution of long-wavelength hydrodynamic modes

is not sufficiently damped, to be ruled by diffusion

(which is a necessary ingredient for the validity of

[19]). It must be stress ed th at th ese num eric al

investigations have strongly revived the interest for

this problem. In particular, they have also stimu-

lated new theoretical efforts for explaining the

power-law divergence of transport coefficients i n

d = 1. One of the main achievements of these

theoretical approaches is that the power-law

divergence turns to a logarithmic one in d = 2,

while the divergence should disappear in d  3.

Despite the difficulty of performing the necessary

large-scale simulations for such systems in d > 1, it

seems that numerics essentially agree with such

predictions.

One can find normal transport properties even

in d = 1, if suitable models are considered. For

instance, momentum conservation can be broken

by adding to the Hamiltonian [1] alocalinterac-

tion potential, U(q

), which breaks translation

invariance, thus restoring finite heat conductivity

(e.g., see Casati et al. 1984). The exce ption to this

case is the harmonic chain with the addition of a

local harmonic potential: in this case the dynamics

is still integrable and there are as many conserved

quantities as degrees of freedom. A further pecu-

liar case is represented by the r otator model in

d = 1, which is known to be nonintegrable. Its

Hamiltonian contains the interaction potential

[1  cos(q

iþ1

 q

)], replacing the algebraic poten-

tials of the FPU chain. Anyway, such a Hamilto-

nian still guarantees momentum conservation,

since the nearest-neighbor form of the interaction

is maintained. Notice that, for small oscillations

around the equilibrium position, also the rotator

potential admits a Taylor-series expansion, whose

first three terms correspond to quadratic, cubic,

and quartic contributions, as in the FPU chain.

Nonetheless, at variance with the FPU problem,

the potential of the rotator model is bounded also

from above. Numerical investigations (Giardina

et al. 2000) have shown that for any finite energy

density and for a sufficiently long finite time,

some previously oscillating rotators start to rotate,

due to local energy fluctuations, that allow to

overtake the potential barrier. These dynamical

configurations typically appear in the form of

spatially localized, synchronous rotating clusters.

Their time evolution is characterized by an

intermittent behavior: they are eventually reab-

sorbed by lattice fluctuations and may reappear

afterwards at other lattice positions. In t his way

they play the role of scattering centers for

hydrodynamic modes. It must be pointed out that

such a qualitative argument is not sufficient for

explaining the onset of a genuine diffusive beha-

vior, compatible with the validity of Fourier’s law.

A hydrodynamic t heory, still to be developed,

could provide a more convincing insight on these

results.

It is worth concluding this section by mentioning

that the overall scenario described above is con-

firmed by numerical studies, relying upon a diff erent

approach, based on equilibrium measurements.

Actually, the linear response theory by Green and

Kubo (see Kubo (1985)) pro vides an alternative, but

essentially equivalent, definition of the thermal

conductivity, according to the expression

 ¼

lim

t!1

lim

N!1

dhJðÞJð0Þi ½21

550 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

The crucial quantity to be comput ed numerically is

the heat-flux time-correlation function C

() =

hJ()J(0)i, where hirepresents the thermodynamic

equilibrium average. In practice, numerical simula-

tions can be performed for a chain of N oscillators

in contact with boundary heat reservoirs at the same

temperature T = T

= T



. The presence of anom-

alous transport coefficients can be singled out by

analyzing the long-time behavior of C

(). It has to

decay at leas t as 

(1þ")

, with ">0 to yield a finite

heat conductivity. In one-dimensional models exhi-

biting the power-law divergence [20] one rather

finds

ðÞ

1þ

½22

where the positive exponent  isthesameappear-

ing in [20] . This relation between space and time

exponents can be easily explained, by considering

that space and time variables depend linearly on

each other t hrough a proportionality constant,

which is the velocity of sound in the lattice. Since

0 <<1, the anomalous behavior observed in

out-of-equilibrium conditions is recovered.

One major problem in perf orming proper numer-

ical studies concerns the control over finite-size

effects, which demands a consistent increase of the

integration time with the system size. This may

yield very extended and expensive c omputations,

mainly when very slow relaxation processes set in.

This is the case of the low-energy regime originally

studied by FPU in their pioneering c omputer

simulations. Numerical analysis indicates that in

this regime the expected behavior of C

(), reported

in eqn [22], sets in after a cros sover time t

,which

increases, for decreasing energy density ,ast

 

2

This seems to be compatible with the studies

described earlier .

We conclude this section by pointing out that this

result also contributes significantly to clarify one of

the basic questions raised by the FPU numerical

experiment.

See also: Dynamical Systems and Thermodynamics;

Ergodic Theory; Fourier Law; Gravitational N-Body

Problem (Classical); Lyapunov Exponents and Strange

Attractors; Nonequilibrium Statistical Mechanics:

Dynamical Systems Approach.

Further Reading

Bocchieri P, Scotti A, Bearzi B, and Loinger A (1970) Anharmonic

chain with Lennard–Jones interaction. Physical Review A 2:

2013.

Bonetto F, Gallavotti G, and Garrido P (1997) Chaotic principle:

an experimental test. Physica D 105: 226.

Casati G, Ford J, Vivaldi F, and Visscher WM (1984) One-

dimensional classical many-body system having a normal

thermal conductivity. Physical Review Letters 52: 1861.

Casetti L, Livi R, and Pettini M (1995) Gaussian model for

chaotic instability of Hamiltonian flows. Physical Review

Letters 74: 375.

Chirikov BV, Izrailev FM, and Tayursky VA (1973) Numerical

experiments on the statistical behaviour of dynamical systems

with a few degrees of freedom. Computational Physics

Communications 5: 11–16.

De Luca J, Lichtenberg AJ, and Lieberman MA (1995)

Time scale to ergodicity in the Fermi–Pasta–Ulam system.

Chao s 5: 283.

Evans DJ, Cohen EGD, and Morriss GP (1993) Probability of

second law violations in shearing steady state. Physical Review

Letters 71: 2401.

Fermi E, Pasta JR, and Ulam S (1965) Studies of nonlinear

problems. In: Collected Works of E. Fermi, vol. 2, p. 978.

Chicago: University of Chicago Press.

Gallavotti G and Cohen EGD (1995a) Dynamical ensembles in

stationary states. Journal of Statistical Physics 80: 931.

Gallavotti G and Cohen EGD (1995b) Dynamical ensembles in

nonequilibrium statistical mechanics. Physical Review Letters

74: 2694.

Giardina´ C, Livi R, Politi A, and Vassalli M (200 0) Finite

thermal conductivity in 1D lattices. Physical Review L etters

84: 2144.

Izrailev FM and Chirikov BV (1966) Statistical properties of a

nonlinear string. Soviet Physics Doklady 11: 30.

Kubo R, Toda M, and Hashitsume N (1985) Statistical Physics II.

Berlin: Springer.

Lepri S, Livi R, and Politi A (1997) Heat conduction in chains of

nonlinear oscillators. Physical Review Letters 78: 1896.

Lepri S, Livi R, and Politi A (1998) Energy transport in

anharmonic lattices close to and far from equilibrium.

Physica D 119: 140.

Lepri S, Livi R, and Politi A (2003) Thermal conduction in

classical low-dimensional lattices. Physics Reports 377: 1.

Livi R, Pettini M, Ruffo S, Sparpaglione M, and Vulpiani A

(1985) Physical Review A 31: 1039, 2740.

Narayan O and Ramaswamy S (2002) Anomalous heat conduc-

tion in one-dimensional momentum-conserving systems. Phy-

sical Review Letters 89: 200601.

Rieder Z, Lebowitz JL, and Lieb E (1967) Properties of a

harmonic crystal in a stationary nonequilibrium state. Journal

of Mathematical Physics 8: 1073.

Shepelyansky D (1997) Low energy chaos in the Fermi–Pasta–

Ulam problem. Nonlinearity 10: 1331.

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations 551

Nonlinear Schro¨ dinger Equations

M J Ablowitz, University of Colorado, Boulder, CO,

USA

B Prinari, Universita

degli Studi di Lecce, Lecce, Italy

Historical Background

Ginzburg–Landau Equations

Nonlinear Schro¨ dinger (NLS) equations have

become one of the most important nonlinear systems

studied in mathematics and physics. Actually, one

can find the essence of NLS equations in the early

work of Ginzburg and Landau (1950) and Ginzburg

(1956) in their study of the macroscopic theory of

superconductivity, and also of Ginzburg and Pitaevskii

(1958), who subsequently investigated the theory of

superfluidity.

By minimizing the free energy of a superconductor

near the superconducting transition, Ginzburg and

Landau arrived at what are now called the

Ginzburg–Landau equatio ns:

ihr



þ  þ j j

¼ 0 ½1

J ¼

ieh



r  r



½

j j

A ½2

where ,  are phenomenological parameters, A the

electromagnetic vector potential, and



denotes

complex conjugate of . The first equation deter-

mines the field based on the applied magnetic

field. The second equation provides the supercon-

ducting current J.

The equation describing the behavior of super-

fluid helium near the transition point in the

stationary case derived in Ginzburg and Pitaevskii

(1958) is completely analogous to eqn [1] in the

phenomenological theory of superconductivity.

Equation [1] contains all the ingredients of the

NLS equations which are discussed below. How-

ever, it was not until the 1960s that the wide

physical importance of NLS equation became

evident. The next section discusses how the NLS

equation historically first appeared in the context of

nonlinear optics.

Nonlinear Optics: Self-Focusing of Optical Beams

in Nonlinear Media

In the mid-1960s, Chiao et al. (1964) and Talanov

(1964) investigated the conditions under which an

electromagnetic beam can produce its own dielectric

waveguide and propagate without spreading. This is

a reflection of the phenomenon of self-focusing. In

fact, self-focusing of optical beams may occur in

materials whose dielectric constant increases with

field intensity. In the general situation, a beam of

uniform intensity in a dielectric broadens due to

diffraction. However, the refractive index of many

physically important materials (the so-called Kerr

materials, such as silica) depends on the field

intensity as follows:

n ¼ n

þ n

jEj

þ

If the term n

jEj

is large enough, the critical angle

for total internal reflection at the beam’s bounda ry

can be greater than the angular divergence due to

diffraction; thus, spreading does not occur as a

result of diffraction. As a consequence, a beam

above a certain critical power level is trapped and

does not spread.

In a remarkable contribution, Kelley (1965)

observed, using computational methods (years

before computational methods became easy to

implement a nd, consequently, so popular) that

when the sel f-foc using e ffect due to the increase in

the nonlinear index is not compensated by diffrac-

tion, there is a buildup in intensity of part of the

beam as a function of the distance in t he direction

of propagation. Consequently, the intensity of the

self-focused regions tended to become ‘‘anoma-

lously large,’’ that is, a singularity appeared to

develop.

Consider as starting equation the electromagne tic

wave equation in the presence of nonlinearities

derived earlier by Chiao et al. (1964):

E 



E 



ðE

EÞ¼0 ½3

where 

jEj

 1. One assumes a linearly polarized

wave of frequency !, propagating along the z-axis,

so that

E ¼

ðEe

iðkz!tÞ

þ c:c:Þ

where c.c. denotes complex conjugation, k = 

1=2

!=c,

the factor exp(ikz  !t) represents the propagating

part, that is, the ‘‘carrier,’’ of the wave, and E is the

slowly varying part. Substituting the above expres-

sion for E into eqn [3], neglecting the third-harmonic

term and the term @

E from r

E (assuming it to be

small), yields

2ik@

Eþ @

þ @



Eþ



jEj

E¼0 ½4

552 Nonlinear Schro¨ dinger Equations

or, with a suitable rescaling of the dependent and

independent variables (E! =((3=4)k



=

)

1=2

z ! 2kz),

þr

þ 2j j

¼ 0 ½5

which is the NLS equation in standard nondimen-

sional form.

It should be remarked here that the name NLS

equation for equations of the form of [5] is natural

due to the formal analogy with the Schro¨ dinger

equation in quantum mechanics:

þr

þ V ¼ 0 ½6

If one sets V = 2j j

in eqn [6], the result is the NLS

equation. In the context of quantum mechanics, a

nonlinear potential arises in the ‘‘mean-field’’

description of interacting particles.

Modifications of [6] also arise as mean-field

descriptions of Bose–Einstein condensates which is

of keen interest in physics (see Pethick and Smith

(2002) and references therein). The normalized

equation is

r

þ Vð x; yÞþ2j j



¼ 0 ½7

where V is an external potential. This is generally

referred to as the Gross–Pitaevskii equation.

Talanov (1965) (see also Zakharov et al. (1971))

investigated the behavior of stationary light beams

in a self-focusing nonlinear medium and found that

for a purely cubic nonlinearity, ‘‘collapse’’ of the

beam can take place. The proof that there is a

singularity in eqn [5] is remarkably straightforward.

This is discussed in the section ‘‘Wave collapse.’’ In

order to avoid wave collapse, other physical effects

(e.g., saturable nonlinearity or dissipation) are

required.

Universal Character of the NLS Equation

It turns out that almost any dispersive, energy-

preserving system gives rise, in an appropriate limit,

to the NLS equation. For instance, one can derive

the NLS from other physically significant equati ons

such as the Klein–Gordon equation

 u

þ u þ ku

¼ 0

and the Korteweg–de Vries (KdV) equation

þ 6uu

þ u

xxx

¼ 0

Actually, the NLS equation provides a ‘‘canonical’’

description for the envelope dynamics of a quasi-

monochromatic plane wave (the carrier wave)

propagating in a weakly nonlinear dispersive med-

ium when dissipative processes are negligible.

Indeed, consider a scalar nonlinear wave equation

written symbolically as

L @

; rðÞu þ GðuÞ¼0

where L is a linear differential operator w ith

constant coefficients and G a nonlinear function

of u and its derivatives. For a real, small-

amplitude solution of magnitude   1, the non-

linear effects can first be neglected, and the

equation admits approximate monochromatic

wave solutions

u ¼  e

iðkx!tÞ

þ c.c. ½8

with small amplitude j j. Substituting [8] into the

linear equation, one can find that the frequency !

and the wave vector k are related by the dispersion

relation

Lði!; ikÞ¼0

Let

! ¼ !ðkÞ

be one of the solutions of the previous equation.

Suppose one is interested in a solution which is

not constant, but slowly varying in space and time.

This has the interpretation of k having a ‘‘sideband’’

wave vector and ! a ‘‘sideband’’ frequency. More

precisely, restricting discussion, for simplici ty, to the

(1 þ 1)-dimensional case, the slowly varying ampli-

tude assumption correspon ds to letting

ðx; tÞ¼ ðX; TÞ¼

iðKxtÞ

where X = x and T = t. Note that K = k and

=! are sometimes referred to as the sideband

wave number and frequency, respectively, because

they correspond to a deviation from the central

wave number k and central frequency !. Looking at

these deviations from the point of view of operators,

whereby ! ! i@

, k !i@

and  ! i@

, K !i@

one has

tot

 ! þ  ¼ ! þ i@

tot

 k þ K ¼ k  i@

Then !(k) can be expanded in a Taylor series

around the central wave number as

! k  i@

ðÞ!ðkÞi!

 

þ

Therefore,

tot

ðkÞ  !ðkÞþi@

½

 !ðkÞi!

 



Nonlinear Schro¨ dinger Equations 553

which shows that, to the leading order,

i

þ !



þ 

¼ 0 ½9

In the moving frame  = X  !

(k)T,  = T  

eqn [9] transforms to









¼ 0

which is the linear Schro¨ dinger equation with the

canonical !

(k)=2 coefficient. On the other hand, if

one considers rather general conservative nonlinear

wave prob lems with leading quadratic or cubic

nonlinearity, asymptotic analysis (e.g., multiple

scale analysis which yields the so-called Stokes–

Poincare´ frequency shift) shows that a wave solution

of the form

uðx; tÞ¼ ðÞe

iðkx!tÞ

þ c:c:

with  = 

t has () satisfying

@

þ nj j

¼ 0 ½10

where the constant coefficient n depends on the

particular equation under study. It should be

remarked here that cubic nonlin earity yields an

O(

) contribution, which is balanced by a slow

timescale of order 

. Putting the linear and non-

linear effects together (i.e., eqns [9] and [10]) implies

that an NLS equation of the form

@

þ nj j

¼ 0

naturally arises. The NLS equation is viewed as a

‘‘universal’’ equation as it generically governs the

slowly varying envelope of a monochromatic wave

train (see also Benney and Newell (1969)).

Physical Applications

The nonlinear propagation of wave packets is

governed by NLS-type systems in several different

branches of scientific and technological applications,

beyond what has been mentioned earlier. Some of

these applications are discussed below.

NLS equation in Water Waves

The NLS equation in the context of small-amplitude

water waves was derived by Zakharov (1968)

(infinite depth) and Benney and Roskes (1969)

(finite depth). The procedure for deriving the NLS

equation from the Euler–Bernoulli equations of fluid

dynamics in one horizontal direction will now be

discussed, under the assumption of small-amplitude

waves and deep water. The interested reader can

also find the details of the derivation in Ablowitz

and Clarkson (2006). The relevant equations are



þ

¼ 0; 1< z <ðx; tÞ½11



¼ 0; z !1 ½12







þ 



þ g ¼ 0; z ¼  ½13



þ 



¼ 

; z ¼  ½14

where  is the velocity potential of an ideal

(i.e., incompressible, irrotational, and invi scid)

fluid, (x, t) is the free surface of the fluid, which

is to be found, in addition to (x, z; t).

Equation [11] expresses the ideal nature of the

fluid; the condition [12] expresses the requirement

that there is no vertical flow at infinity; and eqn [13]

is the Bernoulli equation of energy conservation.

Finally, eqn [14] is a kinematic condition stating

that no flow occurs transverse to the free surface.

At the free boundary, for small amplitudes, one

can expand  = (t, x, ) for   1as

 ¼ ðt; x; 0Þþ

ðt; x; 0Þþ

ðÞ



ðt; x; 0Þþ

and similarly for the derivatives. Second, one

introduces slow temporal and spatial scales (one

expects the slowly varying envelope of the wave to

depend on slow variable s X = x, Z = z, T = t).

Finally, because of the quadratic nonlinearity one

expects second harmonics to be generated; hence,

 ¼ Ae

iþjkjz

þ c:c:



þ  A

2iþ2jkjz

þ c:c: þ







 ¼ Be

i

þ c:c:



þ  B

2i

þ c:c: þ







where A, A



 depend on X, Z, T and B, B





depend on X, T (



 and



 are mean contributions,

which are real) and =kx  !t with the dispersion

relation !

= gjkj . Substituting this ansatz into the

equations, one obtains from the order-

terms

2i!A







¼ 0 ½15

where v

= !

(k) = g=2! is the group velocity and the

new variables  = T,  = X  v

Equation [15] is the typical formulation of the

(1 þ 1)-dimensional NLS equation found in water

wave theory for large depth.

In the section ‘‘NLS in nonlinear optics,’’ a

special solution to (a rescaled version of) eqn [15],

namely a soliton solution, is discussed in the

554 Nonlinear Schro¨ dinger Equations

context of nonlinear optics. It should be

remarked here that the coefficients of both terms



and jAj

A have the same sign. This is necessary

for a decaying soliton solution to exist (see, e.g.,

Lighthill (1965)).

NLS in Nonlinear Optics

The NLS equation also describes self-compression

and self-modulation of electromagnetic wave pack-

ets in weakly nonlinear media. Hasegawa and

Tappert (1973a, b) first deri ved the NLS equation

in the context of fiber optics. Light-wave propaga-

tion in a fibe r is mainly affected by: (1) group

velocity dispersion (GVD), that is, the frequency

dependence of the group velocity originating from

the refractive index of the fiber and (2) fiber

nonlinearity (the so-called Kerr effect), originating

from the dependence of the refractive index on the

intensity of the optical pulse. In the presence of

GVD and Kerr nonlinearity, the refractive index is

expressed as

nð!; EÞ¼n

ð!Þþn

jEj

½16

where ! and E represent the frequency and

electric field of the light wave, respectively, n

(!)

is the frequency-dependent linear refractive index,

and the constant n

,referredtoastheKerr

coefficient, is ‘‘small’’ but can have significant

impact since the nonlinear effects accumulate over

long distances. Normally, the electric field is

modulated into a slowly varying amplitude of a

carrier wave:

Eðz; tÞ¼Eðz; tÞ e

iðk

z!

tÞ

þ c:c: ½ 17

where z denotes the distance along the fiber, t the

time, k

= k

) the wave number, !

the fre-

quency, and E(z, t) the envelope of the electromag-

netic field.

A Taylor series expansion of the dispersion

relation (see also the section ‘‘Universal character

of the NLS equation’’)

kð!; EÞ¼

ðn

ð!Þþn

jEj

around the carrier frequency ! = !

yields

k  k

¼ k

ð!

Þð!  !

Þþ

ð!

ð!  !

jEj

½18

where the prime represents derivative with respect to

! and k

= k(!

). Replacing k  k

and !  !

their Fourier operator equivalents, i@

and i@

resp.,

using k  k

= (!=c)n

(!) and letting eqn [18]

operate on E yields

þ k

ð!





ð!

þ jEj

E¼0 ½19

where  = !

=cA

eff

, with A

eff

being the effective

cross-section area of the fiber (the factor 1=A

eff

comes from a more detailed derivati on which takes

into accou nt the finite size of the fiber; the factor

1=A

eff

is needed in order to account for the variati on

of field intensity in the cross section of the fiber).

Note that k

) = 1=v

, where v

represents the

group velocity of the wave train. Introducing dimen-

sionless variables t

= t

ret



, z

= z=z



, q = E=

ﬃﬃﬃﬃﬃ



yields the NLS equation

sgnðk

ð!

ÞÞ

þjqj

q ¼ 0 ½20

where t



, P



are the characteristic time and power,

respectively, and t

ret

= t  k

)z = t  z=v

, z



1=P



, with the constraint that the ‘‘nonlinear

length’’ is balanced by the linear dispersion time,

that is, t



= ðz



jk

)jÞ

1=2

There are two cases of physical interest depending

on the sign of k

. The so-called focusing case occurs

when k

< 0; this is called ‘‘anomalous’’ dispersion.

The defocusing case obtains when the dispersion is

‘‘normal’’: k

> 0.

Now write eqn [20] in the form

þ q

 2jqj

q ¼ 0 ½21

with  corresponding to the focusing (þ) and

defocusing () case, respectively. The focusing NLS

equation admits special solutions called ‘‘bright’’

solitons (solutions that are traveling localized

‘‘humps’’). A pure one-soliton solution in the

focusing (þ) case has the form

qx; tðÞ¼ sech  x þ 2t  x

ðÞ½e

i

½22

where =x þ (

 

)t þ 

. The parameters 

and  are such that  = =2 þ i=2 is an eigenvalue

from the inverse scattering transform analysis.

The defocusing () NLS equation does not admit

solitons that decay at infinity. However, it does admit

soliton solutions which have a nontrivial background

intensity (called ‘‘dark’’ and ‘‘gray’’ solitons). A dark-

soliton solution has the form

qðx; tÞ¼ tanh xðÞe

2i

½23

Note that q ! as x !1. A gray-soliton

solution is

qðx; tÞ¼ 1  B

sech

Bx x

ðÞðÞ

1=2

i x;tðÞ

½24

Nonlinear Schro¨ dinger Equations 555

with

 x; tðÞ¼

2  B



t þ 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  B

þ tan

1

B tanh BxðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  B



þ 

and B

< 1. Note that as B ! 1



, the gray soliton

becomes a dark soliton, taking 

= =2.

Recall that the solutions [23] and [24] can be

allowed to travel uniformly by making a Galilean

transformation, that is, taking into account that if

(x, t) is a solution of [21], then so is

ðx; tÞ¼q

ðx  vt; tÞe

i kx!tðÞ

with k = v and ! = k

=2.

It should also be remarked that Ablowitz et al.

(1997) have shown that, in quadratically nonlinear

optical materials, more complicated NLS-type equa-

tions arise. These equations are analogous to the

finite-depth multi dimensional nonlocal NLS-type

systems derived in the context of water waves by

Benney and Roskes (1967) and later by Davey and

Stewartson (1974).

Optical Communications

Hasegawa and Tappert (1973) first suggested using

solitons as the ‘‘bit’’ format for transmission of

information in optical fiber systems. Motivated by

this, in 1980, scientists at Bell Laboratories observed

solitons (described by the NLS equation) in optical

fibers ( Mollenauer et al. 1980 ). The development of

optical amplifiers (erbium-doped amplifiers) in the

mid-1980s provided a mechanism to compensate

fiber loss, and this permitted the transmission of

information entirely optically over long distances.

With damping and amplification included (see, e.g.,

Hasegawa and Kodama (1995)), the NLS equation

[20] takes the form

sgnðk

ð!

ÞÞ

þ gðzÞjqj

q ¼ 0 ½25

where g(z) = a

exp( 2z=z

), 0 < z < z

,andperi-

odically extended thereafter, and a

is determined by

< g >¼

gðz=z

Þdz ¼ 1

with z

= l



, l

being the amplifier length.

Remarkably, asymptotic analysis (z

 1) shows

that, to leading order, q(z, t) still satisfies the NLS

equation [20].

Amplifiers, however, introduce small amounts of

noise to the system, which causes the temporal

position of the soliton to fluctuate (cf. Gordon and

Haus (1986)) and thus limits the distance signals can

be reliably transmitted to. Soliton control mechan-

isms were introduced in the early 1990s in order to

deal with these difficulties (cf. Mecozzi et al. (1991)

and Kodama and Hasegawa (1992)).

By the mid-1990s, the development of all optical

transmission syste ms began to take great advantage

of wavelength-division-multiplexing (WDM), that

is, the simultaneous transmission of multiple

signals in different frequency (or equivalently

wavelength) ‘‘channels’’ (Hasegawa 2000). How-

ever, it was found that a serious problem affected

WDM systems. Namely, the interactions of soli-

tons traveling at different velocitie s cause resonant

amplifier-induced instabilities in adjacent fre-

quency channels (four-wave mixing ( Mamyshev

and Mollenauer 1996, Ablowitz et al. 1996)). In

order to avoid these instabilities, researchers

developed and analyzed dispersion-manage d (DM)

transmission sys tems (cf. Hasegawa (2000)). In a

DM transmission system, the fiber is composed of

alternating sections of positive (normal) and

negative (anomalous) dispersion fibers. The

(dimensionless) NLS equation that g overns this

phenomenon is

dðzÞ

þ gðzÞjqj

q ¼ 0 ½26

where d(z)isusuallytakentobeaperiodic,large,

rapidly varying function of th e form d(z) = 

(z), with j(z)j1 and having zero average in

the period z

(generally the same as that of the

amplifier). In fact, asymptotic analysis of [26]

yields a nonlocal NLS-type equation (Gabitov and

Turitsyn 1996, Ablowitz and Biondini 1998). It has

also been shown tha t eqn [26] admits various types

of optical pulse s, such as DM solitons (Ablowitz

and Biondini 1998), and quasilinear modes (Ablowitz

et al. 2001).

NLS Equation in Other Settings

Many other interesting applications of the NLS

equations exist in such different areas of physics as

magnetic spin waves (see, e.g., the work by Zvezdin

and Popkov (1983) andalsobyKalinikos et al.

(1997)), plasma physics (cf. the work by Zakharov

(1972) on collapse of Langmuir w aves), other areas

of fluid dynamics, etc. (the interested reader can

find an overview in the monograph by Ablowitz

(1981)).

Mathematical Framework

Mathematically, the NLS equation had attained

broad significance since it is integrable via

556 Nonlinear Schro¨ dinger Equations