# Nonlinear Schrodinger systems: continuous and discrete

The **Nonlinear Schrodinger (NLS) equation** is a prototypical
dispersive nonlinear partial differential equation (PDE) that has
been derived in many areas of physics and analyzed mathematically
for over 40 years. Historically the essence of NLS equations can be
found 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. Nonetheless,
it was not until the works of Chiao *et al* (1964) and
Talanov (1964) that the wider physical importance of NLS equation
became evident, especially in connection with the phenomenon of
self-focusing and the conditions under which an electromagnetic beam
can propagate without spreading in nonlinear media. In the general
situation, an optical beam in a dielectric broadens due to
diffraction. However, in materials whose dielectric constant
increases with the field intensity, the critical angle for internal
reflection at the beam's boundary can become greater than the
angular divergence due to diffraction and as a consequence the beam
does not spread and can, in some situations, continue to focus into
extremely high intensity spots.

Starting from the electromagnetic wave equation in the presence of nonlinearities and assuming a linearly polarized wave propagating along the \(z\)-axis, after a suitable rescaling of the dependent and independent variables one can derive for the propagation of the electromagnetic field the NLS equation in standard nondimensional form \[ i\partial_{z}\psi+\Delta_{\perp} \psi+2\left| \psi\right|^2 \psi=0 \] where \(\psi\) is proportional to the slowly varying complex envelope of the electromagnetic field, \(z\) is the propagation variable, and \(\Delta_{\perp}\) denotes the Laplacian with respect to the transverse coordinates. Subscripts \(x,y,z,t\) will denote partial differentiation throughout this entry.

Beside the fact that NLS systems have direct applications in many physical problems, the importance of the NLS equation is also due to its universal character (cf. Benney and Newell, 1967). Generically speaking, most weakly nonlinear, dispersive, energy-preserving systems give rise, in an appropriate limit, to the NLS equation. Specifically, the NLS equation provides a "canonical" description for the envelope dynamics of a quasi-monocromatic plane wave propagating in a weakly nonlinear dispersive medium when dissipation can be neglected.

Mathematically, the NLS equation attains broad significance since, in one transverse dimension, it is integrable via the Inverse Scattering Transform (IST, for brevity) -- which is a nonlinear Fourier Transform -- it admits multisoliton solutions, it has an infinite number of conserved quantities, and it possesses many other interesting properties.

There has been a vast amount of literature involving the NLS
equation over the years, but recently there has been additional
interest, mainly due to the developments in nonlinear optics and
soft-condensed matter physics. In the optical context, the
experimental developments involving localized pulses in arrays of
coupled optical waveguides (cf. Eisenberg *et al*, 1998) have drawn
attention to discrete NLS models (where the fields are substituted
by appropriate finite differences). Related problems involving NLS
equations on a lattice background (cf. Efremidis *et al*, 2003) have
also generated considerable interest. The vector generalization of
the NLS equation has also proved to be particularly valuable from
the point of view of nonlinear optics. On the other hand, the
experimental realization of Bose-Einstein condensates (BECs) and
their mean field modeling by the so-called Gross-Pitaevskii (cf.
Pethick and Smith, 2002) equation which, like optical pulses on a
lattice background, is an NLS equation with an external potential,
has opened new avenues for the study of NLS-type equations.

The following sections elucidate some of the physical and the mathematical aspects of NLS systems, both continuous and discrete, scalar and vector, in one or more spatial dimensions.

## Continuous models

### Scalar (1+1)-dimensional systems

The nonlinear propagation of wave packets is governed by NLS-type
systems in such diverse fields as fluid dynamics (cf. Ablowitz and
Segur, 1981), nonlinear optics (cf. Agrawal, 2001), magnetic spin
waves (cf. Zvedzin and Popkov, 1983, Chen *et al*, 1994),
plasma physics (Zakharov 1972) etc.

For example, the NLS equation describes self-compression and self-modulation of electromagnetic wave packets in weakly nonlinear media. Hasegawa and Tappert (1973a, b) first derived the NLS equation in fiber optics, taking into account both dispersion and nonlinearity. Detailed derivations can be found in texts [cf. Hasegawa and Kodama, 1995 and refs therein].

Dispersion originates from the frequency dependence of the refractive index of the fiber and leads to frequency dependence of the group velocity; this is usually called group velocity dispersion or simply GVD. Due to GVD, different spectral components of an optical pulse propagate at different group velocities and thus arrive at different times. This leads to pulse broadening, resulting in signal distortion.

Fiber nonlinearity is due to the so-called Kerr effect, i.e. the dependence of the refractive index on the intensity of the optical pulse. In the presence of GVD and Kerr nonlinearity, and neglecting polarization-dependent effects, the refractive index is expressed as \[ n(\omega,E)=n_0(\omega)+n_2|E|^2 \] where \(\omega\) and \(E\) represent the frequency and electric field of the lightwave, \(n_0(\omega)\) is the frequency dependent linear refractive index and the constant \(n_2\ ,\) the Kerr coefficient, has a value of approximately \(10^{-22}m^2/W\ .\) Even though fiber nonlinearity is small, the nonlinear effects accumulate over long distances and can have a significant impact due to the high intensity of the lightwave over the small fiber cross section. By itself, the Kerr nonlinearity produces as intensity dependent phase shift which results is spectral broadening during propagation.

In the usual transmission process with lightwaves, the electric field is modulated into a slowly varying amplitude of a carrier wave. Concretely, a modulated electromagnetic lightwave is written as \[ E(z,t)=\epsilon(z,t)e^{i(k_0z-\omega_0t)}+c.c. \] where c.c. denotes complex conjugation, \(z\) the distance along the fiber, \(t\) the time, \(k_0=k_0(\omega_0)\) the wavenumber, \(\omega_0\) the frequency and \(\epsilon(z,t)\) is the complex envelope of the electromagnetic field.

A simplified derivation can be conveniently obtained from the nonlinear dispersion relation: \[ k(\omega,E)=\frac{\omega}{c}(n_0(\omega)+n_2|E|^2) \] where \(c\) denotes the speed of light.

A Taylor series expansion of \(k(\omega,E)\) around the carrier frequency \(\omega=\omega_0\) yields \[\tag{1} k-k_0=k'(\omega_0)(\omega-\omega_0)+\frac{k''(\omega_0)}2(\omega-\omega_0)^2+\frac{\omega_0n_2}{c}|E|^2 \]

where \('\) represents derivative with respect to \(\omega\) and \(k_0=k(\omega_0)\ .\) Replacing \(k-k_0\) and \(\omega-\omega_0\) by their Fourier operator equivalents \(i\partial/\partial z\) and \(i\partial/\partial t\) resp., using \(k-k_0=\frac{\omega}{c}n_0(\omega)\) and letting Eq. (1) operate on \(\epsilon\) yields \[ i\left(\frac{\partial \epsilon}{\partial z} +k_0'(\omega_0)\frac{\partial \epsilon}{\partial t}\right) -\frac{k_0''(\omega_0)}2 \frac{\partial^2 \epsilon}{\partial t^2}+\nu|\epsilon|^2\epsilon=0 \] where \(\nu=\frac{\omega_0n_2}{3cA_{eff}}\ ,\) with \(A_{eff}\) being the effective cross section area of the fiber (the factor \(1/A_{eff}\) comes from a more detailed derivation that takes into account the transverse dimensions; it is used in order to take into account the variation of field intensity in the cross section of fiber). Note that \(k_0'(\omega_0)=1/v_g\) where \(v_g\) represents the group velocity of the wavetrain.

In order to obtain a dimensionless equation, it is standard to introduce a retarded time coordinate \(t_{ret}=t-k_0'(\omega_0)z\) and dimensionless variables \(t'=t_{ret}/t_*\ ,\) \(z'=z/z_*\) and \(q=\epsilon/\sqrt{P_*}\) where \(t_*,z_*,P_*\) are the characteristic time, distance and power respectively. Substitution of this coordinate transformation and choosing the dimensionless variables as \(z_*=1/ \nu P_*\ ,\) \(t_*^2=z_*|-k''(\omega_0)|\) [and for convenience dropping the \('\)] yields the NLS equation \[\tag{2} i\frac{\partial q}{\partial z}+\frac{\mathrm{sgn}(-k_0''(\omega_0))}2 \frac{\partial^2 q}{\partial t^2}+|q|^2q=0. \]

There are two cases of physical interest depending on the sign of \(-k_0''\ .\)

Replacing \(z\to2\mathrm{sgn}(-k_0''(\omega_0))z\) the NLS equation in "standard" form is given by \[\tag{3} iq_{z}+q_{tt}\pm2\left| q\right|^2q=0. \]

In these notations, the focusing case is given by the (+) sign in Eq.(3), and it corresponds to anomalous dispersion. The defocusing case obtains when the dispersion is normal, and it corresponds to the (-) sign in Eq.(3).

The NLS equation possesses soliton solutions, which are exact solutions decaying to a background state. The focusing (+) NLS equation admits so-called "bright" solitons (namely, solutions that are localized travelling "humps"). A pure one-soliton solution of the focusing NLS has the form \[ q(z,t)=\eta\ sech\ \left[ \eta\left( t+2\xi z-t_{0}\right) \right] e^{-i\theta(z,t)} \] where \(\theta(z,t)=\xi t+\left( \xi^2-\eta^2\right) z+\theta_{0}\ .\) A typical bright soliton is depicted in Fig. 1 (with \(\eta=1/\sqrt2\)).

It is worth noting that in nonlinear optics and many other areas of physics solitary waves are usually called solitons, despite the fact that they generally do not interact elastically. Indeed today most physicists and engineers use the word soliton in this broader sense.

The defocusing (-) NLS equation does not admit solitons that vanish at infinity. However, it does admit soliton solutions on a nontrivial background, called "dark" and "gray" solitons. A dark soliton is a solution of the form \[\tag{4} q(z,t)=q_0\tanh\left( q_0 t\right) e^{2iq_0^2z}\, . \]

A gray soliton solution is \[\tag{5} q(z,t)= q_0e^{2iq_0^2 z}\left[\, \cos\alpha + i\sin\alpha\,\tanh\left[\sin\alpha\,q_0(t-2q_0\cos\alpha\,z-t_0)\right] \,\right] \]

with \(q_0\ ,\) \(\alpha\) and \(t_0\) arbitrary real parameters. Such solutions satisfy the boundary conditions \[ q(z,t)\to q_\pm(z)=q_0e^{2iq_0^2 z \pm i\alpha} \qquad \text{as}\quad x\to\pm\infty \] and appear as localized dips of intensity \(q_0^2\sin^2\alpha\) on the background field \(q_0\ .\) As \(\cos \alpha \rightarrow 0^{+}\ ,\) the gray soliton becomes a dark soliton, which is stationary. In Fig. 2 we illustrate a typical gray soliton (with \(\alpha= \pi/3\)).

Importantly, the solution of the NLS equation for both decaying initial data and for data which tend to constant amplitude at infinity were obtained by the method of the Inverse Scattering Transform (see below for a brief description) by Zakharov and Shabat (1972, 1973).

One of the most remarkable properties of soliton solutions is that interacting scalar solitons affect each other only by a phase shift, that depends only on the soliton powers and velocities, both of which are conserved quantities. Thus, when two soliton collisions occur sequentially, the outcome of the first collision does not affect the second collision, except for a uniform phase shift.

In the context of small-amplitude water waves, the NLS equation was derived by Zakharov (1968) for the case of infinite depth and Benney and Roskes (1969) for finite depth (see also the discussion below regarding the difference between standard NLS equations and Benney-Roskes/Davey-Stewartson type equations). Basically, the NLS equation is obtained from the Euler-Bernoulli equations for the dynamics of an ideal (i.e., incompressible, irrotational and inviscid) fluid under the assumption of a small amplitude quasi-monochromatic wave expansion.

Finally, it should also be mentioned that Ablowitz *et al*
(1997,2001) have shown that, in quadratically nonlinear optical
materials, more complicated NLS-type equations can arise.

### Vector (1+1)-dimensional systems

In many applications, vector NLS (VNLS) systems are the key governing equations. Physically, the VNLS arises under conditions similar to those described by NLS whenever there are suitable multiple wavetrains moving with nearly the same group velocity (Roskes 1976). Moreover, VNLS also models systems where the electromagnetic field has more than one component. For example, in optical fibers and waveguides, the propagating electric field has two polarized components transverse to the direction of propagation. The dimensionless system \[\tag{6} \begin{matrix} iu_{z}+\frac12u_{tt}+\left( \left| u\right|^2+ \left| v\right|^2\right) u =0 \\ iv_{z} +\frac12v_{tt}+\left( \left| u\right| ^2+\left| v\right|^2\right) v =0 \end{matrix} \]

was considered by Manakov (1974) as an asymptotic model governing the propagation of the electric field in a waveguide, where \(z\) is the normalized distance along the waveguide, \(t\) is a transverse coordinate and \((u,v)^T\) (the superscript \(T\) denotes matrix transpose) are the transverse components of the complex electromagnetic field envelope. Manakov was able to integrate the above VNLS system by the IST method.

Subsequently, Menyuk (1987) showed that in optical fibers with constant birefringence, the two polarization components \(\left(u,v\right)^T\) of the complex electromagnetic field envelope orthogonal to direction of propagation along a fiber satisfy asymptotically the following nondimensional equations \[ \begin{matrix} i\left( u_{z}+\delta u_{t}\right) +\frac{d}2u_{tt}+\left( \left| u\right|^2+\alpha\left| v\right|^2\right) u =0 \\ i\left( v_{z}-\delta v_{t}\right) +\frac{d}2v_{tt}+\left( \alpha \left| u\right|^2+\left| v\right|^2\right) v =0 \end{matrix} \] where \(\delta\) represents the group velocity "mismatch" between the components \(u\) and \(v\ ,\) \(d\) is the group velocity dispersion and \(\alpha\) is a constant depending on the polarization properties of the fiber. The physical phenomenon of birefringence implies that the phase and group velocities of the electromagnetic wave are different for each polarization component. It is important to realize, however, that the derivation of the above equations assumes that certain nonlinear (four-wave mixing) terms are neglected. In the general case, i.e. when \(\alpha\ne1\ ,\) the vector NLS system is unlikely to be integrable. However, in a communications environment, due to the distances involved, not only does the birefringence evolve, but it does so randomly and on a scale much faster than the distances required for communication transmission. In this case, Menyuk (1999) showed, after averaging over the fast birefringence fluctuations, the relevant equation is the above but with \(\alpha=1\) and \(\delta=0\) -- that is, it reduces to the integrable VNLS derived by Manakov, which therefore attains broader relevance.

As indicated above, the Manakov system (6) is
integrable, and it possesses vector soliton solutions.
In the focusing case -- that is, with a plus sign in front of the
cubic nonlinear terms -- these are bright solitons whose shape is
the same as that of the bright solitons of the scalar NLS equation,
multiplied by a constant polarization vector.
Unlike scalar solitons, however, the collision of solitons with
internal degrees of freedom
(e.g. vector or matrix solitons) can be highly nontrivial:
even though the collision is elastic, in the sense that the total energy
of each soliton is conserved, there can be a significant
redistribution of energy among the components. It has been shown by
Soljacic *et al* (1998) that the parameters
controlling the energy switching between components exhibit
nontrivial transformation of information. This set forth the
experimental foundations of computation with solitons. Despite the
vector nature of the problem, one can show that the multisoliton
interaction process is nevertheless pair-wise and the net result of
the interaction is independent of the order in which such collisions
occur. This interaction property can be related to the fact that the
map determining the interaction of two solitons satisfies the
Yang-Baxter relation (Ablowitz *et al*, 2004b).

The defocusing VNLS equation -- namely, Eqs. (6)
with a minus sign in front of the nonlinear terms --
admits "dark-dark soliton" solutions; i.e.,
solitons which have dark solitonic behavior in both components, as
well as "dark-bright" soliton solutions, which contain one dark
and one bright component (cf. Kivshar and Turitsyn, 1993). Although
the mathematical properties of VNLS have been investigated for
decades, the IST for the vector system
under nonvanishing boundary conditions has been developed only
recently; e.g., see Prinari *et al* (2006).

### Scalar multidimensional systems

The NLS equation in 2 spatial dimensions, i.e. \[\tag{7} i\psi_{t}+\Delta\psi+\left| \psi\right|^2\psi=0,\qquad \mathbf{x}=(x,y) \in\mathbb{R}^2 \]

has been investigated shortly after the early studies on the one-dimensional
equation. Note that in optics the transverse Laplacian, here simply indicated by
\(\Delta\ ,\) describes wave
diffraction. Remarkable early direct numerical simulations and scaling arguments
by Kelley (1965) indicated wave collapse could occur.
Vlasov *et al* (1971) showed that for a purely cubic nonlinearity in a self-focusing nonlinear medium, the
phenomenon of wave collapse takes place and the light beam blows up
in a finite time.
The proof that a finite-time singularity can occur in
Eq.(7) is remarkably straightforward (Vlasov *et al* 1971) and it is based on the virial
theorem (see also Ablowitz and Segur, 1979).
One can also prove rigorously (cf. C Sulem and P L Sulem 1999)
that, for initial conditions for which the
Hamiltonian \( H=\int\left(\left| \nabla\psi\right|^2
- (1/2)\left|\psi\right|^4\right)d\mathbf{x} \) is negative,
there exists a time \(t_*\) such that the quantity
\[
\int\left| \nabla\psi\right|^2d\mathbf{x}
\]
becomes infinite as \(t\) approaches \(t_*\ ,\) which in turn implies that
\(\psi\) also becomes infinite as \(t\rightarrow t_*\) (blowup in finite time).
It is worth mentioning that near blowup the solution displays universal scaling properties.

Results are also available for the more general NLS equation in \(d\) spatial dimensions and with generic power nonlinearity: \[ i\psi_{t}+\Delta_{d}\psi+\left| \psi\right|^{2\sigma}\psi=0, \qquad\mathbf{x}\in\mathbb{R}^{d}, \] where \(\Delta_{d}\) is the \(d\)-dimensional Laplacian. More precisely, one has the following cases:

- Supercritical (\(\sigma d>2\)): the solution blows up.
- Critical (\(\sigma d=2\)): blowup can occur or global solution can exist.
- Subcritical (\(\sigma d<2\)): global solutions exist.

The first proof of global existence of solutions to the
focusing NLS equation in the sub-critical dimension was given by Ginibre
and Velo (1979). There are many references to this interesting subject; see for example
Papanicolau *et al* (1994), C Sulem and PL Sulem (1999),
Merle and Raphael (2004) and references therein.

Finally, we mention the Zakharov system for Langmuir turbulence in plasmas, where the mean field obeys a dynamical equation (cf. Zakharov, 1972).

### Benney-Roskes and Davey-Stewartson models

Yet another example of the wide applicability of NLS-type systems is
given by nonlocal nonlinear Schrodinger systems, where the NLS-type field
is coupled to an underlying mean term.
The underlying governing systems of equations typically have
quadratic nonlinearities; this happens for example in water waves
and \(\chi^{(2)}\) nonlinear-optical media.
Such systems -- which we refer to here as *NLS with mean terms*,
or NLSM for brevity --
are also referred to as Benney-Roskes or Davey-Stewartson type.
The key difference in the derivation of such systems from that of
the usual NLS equation is the presence of an additional mean term
that corresponds to the zero-th harmonic of the
field at the fundamental frequency. This leads to a system of
equations that describes the nonlocal-nonlinear coupling between a
dynamic field that is associated with the first harmonic (with a
"cascaded" effect from the second harmonic), and a static field
that is associated with the mean term (i.e., the zero-th harmonic).
In water waves, with finite depth, the NLSM equation was first
derived by Benney and Roskes in 1969.
It was re-derived a few years later by Davey and Stewartson (1974),
who put it in a convenient form:
\[\tag{8}
\begin{matrix}
iu_{z}+ \frac12(\sigma_1u_{xx}+u_{yy})+ \sigma_2\left| u\right|^2u-\rho u\phi_{x}=0
\\
\phi_{xx}+\nu\phi_{yy}=\left( \left| u\right|^2\right)_{x}\qquad
\end{matrix}
\]

where \(\sigma_1,\sigma_2,\nu\) and \(\rho\) are real; \(u\) is proportional to the amplitude of the first harmonic and \(v\) the mean (zeroth harmonic) in the slowly varying envelope expansion. The system reduces to the classical NLS equation when \(\rho=0\ ,\) because in that case the mean field \(\phi\) does not couple to the harmonic field \(u\ .\) In addition, when \(\nu=0\) the first equation reduces to a classical NLS with the cubic term \((\sigma_2-\rho)\left| u\right|^2u\ .\) In optics \(\rho>0\ ,\) whereas in water waves \(\rho<0\ .\) In either case, the NLSM system is a nonlocal system of equations; analysis and numerical simulations (cf. Ablowitz, Bakirtas and Ilan, 2005) indicate that when \(\sigma_1,\sigma_2, \nu>0\) collapse can occur in the NLSM equations in the same spirit as in NLS collapse. Near the collapse region the solution asymptotically attains the form of a special localized wave -which satisfies a stationary system of equations, obtained by setting \(u=e^{i\lambda z}F(x,y), \phi=G(x,y)\ ,\) \(\lambda\) constant.

Interestingly, when \(\sigma_1,\sigma_2\) are \(\pm1\ ,\) the above (2+1)-dimensional equations are integrable by IST; cf. Ablowitz and Clarkson, 1991. This limit corresponds to the shallow water wave limit of the Benney-Roskes/Davey-Stewartson system.

### Vector multidimensional systems

Probably the best known multidimensional NLS system is the extension of the 1+1 vector NLS dimensional equation described above in optics, taking into account transverse variations. In the multidimensional case it takes the form \[ \begin{matrix} i\left( u_{z}+\delta u_{t}\right) +\frac{d}2u_{tt}+ \frac12\Delta u + \left( \left| u\right|^2+\alpha\left| v\right|^2\right) u =0 \\ i\left( v_{z}-\delta v_{t}\right) +\frac{d}2v_{tt}+ \frac12\Delta u + \left( \alpha \left| u\right| ^2+\left| v\right|^2\right) v=0 \end{matrix} \] where \(\delta\) represents the group velocity "mismatch" between the transverse components \((u,v)^T\) of the electromagnetic field envelope, \(d\) is the dimensionless dispersion coefficient, \(\alpha\) is a constant depending on the polarization properties of the medium and \(\Delta\) is the two-dimensional Laplacian in the directions transverse to the direction of propagation \(z\ .\)

Ablowitz, Biondini and Blair (1997,2001) also derived vector extensions of the NLSM equations in electromagnetics from Maxwell's equations in \(\chi^{(2)}\) nonlinear-optical media. These equations reduce to the above system when the \(\chi^{(2)}\) contributions vanish. The vector NLSM system takes the following form \[ \begin{matrix} i\left( u_{j,z}+\delta_j u_{j,t}\right) + \frac12(\Delta u_j+ d_{1,j}u_{j,tt}) + \left( M_{1,j}\left| u_j\right|^2+ M_{2,\bar{j}}\left| u_{\bar{j}}\right|^2 + M_{3,j}\phi_j+ M_{4,\bar{j}}\phi_{\bar{j}} \right) u_j =0\qquad \\ \phi_{j,xx}+s_{1,j}\phi_{j,yy}+s_{2,j}\phi_{j,tt} + s_{3,\bar{j}}\phi_{\bar{j},xy} + \left( N_{1,j}\partial^2_{t}+N_{2,j}\partial^2_{x}+N_{3,j}\partial^2_{y}+N_{4,j}\partial^2_{xy}\right)\left(\left| u_j\right|^2- \left| u_{\bar{j}}\right|^2\right) =0 \end{matrix} \] where \(j=1,2;\bar{j}=2,1, \delta_j\) represents the group velocity "mismatch", \(d_{1,j},s_{k,j}\) are coefficients related to linear dispersion relations and \(M_{k,j},N_{k,j}\) depend on the nonlinear coefficients\[\chi^{(2)},\chi^{(3)}\ .\]

## Discrete models

### Discrete (1+1)-dimensional systems

Both the NLS and the VNLS equations admit integrable and nonintegrable discretizations which, besides being used as numerical schemes for the continuous counterparts, also have physical applications as discrete systems.

An important discretization of NLS is the following \[\tag{9} i\frac{d}{dt}q_{n}=\frac1{h^2}\left(q_{n+1}-2q_{n}+q_{n-1}\right) \pm\left| q_{n}\right|^2\left( q_{n+1}+q_{n-1}\right) \]

which is referred to here as the integrable discrete NLS (IDNLS) or also sometimes in the literature as the Ablowitz-Ladik equation. It is an \(O(h^2)\) finite-difference approximation of NLS which is integrable via the IST and has soliton solutions on the infinite lattice (Ablowitz and Ladik 1975, 1976).

If the nonlinear term in the IDNLS is changed to \(2\left| q_{n}\right|^2q_{n}\ ,\) one obtains an equation which is often called the Discrete Nonlinear Schrodinger (DNLS) equation. Since there are regimes where DNLS exhibits chaotic dynamics (cf. Ablowitz and Clarkson, 1991) it is likely to be not integrable.

The DNLS equation describes a simple model for a lattice of coupled anharmonic oscillators. In one spatial dimension, the equation in its simplest form is \[\tag{10} i\frac{d}{dz}\psi_{n}+\psi_{n+1}+\psi_{n-1}+\gamma\left|\psi_{n}\right|^2\psi_{n} =0 \]

where \(\psi_{n}\) is the complex mode amplitude of the oscillator at site \(n\) and \(\gamma\) is an anharmonic parameter, and it is a standard finite difference approximation to the NLS equation. The prototypical application of DNLS is given by coupled nonlinear optical waveguides "etched" into a suitable optical material and well separated from each other in, say, the \(x\)-direction (or \(n\)-direction), with propagation occurring in the longitudinal, direction \(z\) (see Fig. 3 which depicts the waveguide structure). Starting from Maxwell's equations, one can model the governing wave equation for the electromagnetic field in the \(x\)-direction in Kerr nonlinear materials in terms of the following nonlinear Helmholtz equation \[ \Psi_{zz}+\Psi_{xx}+\left( f(x)+\delta\left| \Psi\right|^2\right)\Psi=0 \] where \(\left| \delta\right| \ll1\) and \(f(x)\) models the linear index of refraction. Then one expands the solution \(\Psi\) in terms of a suitable series of functions \[ \Psi=\sum_{m=-\infty}^{\infty}E_{m}(\delta z)\psi(x-md)e^{-i\lambda_{0}z} \] where \(d\) is the spacing of the waveguide array, and \(\psi_{m}=\psi(x-md)\) has one bound state \(\lambda_{0}\ .\) Assuming the eigenfunctions to be localized corresponding to waveguides that are well separated and imposing the condition of maximal balance, the resulting equation for \(E_{n}\) turns out to be the DNLS equation (Ablowitz and Musslimani (2003c)).

The DNLS equation was first derived in the context of nonlinear
optics by Christodoulides and Joseph (1988). The equation had
previously been studied by Davydov (1973) in molecular biology and
Su *et al* (1979) in condensed matter physics.
The paper by Eilbeck *et al* (1985) was the first
to point out the universal nature of the discrete NLS equation, and reported a
number of applications, especially for periodic solutions with small periods.
Experimentally, discrete solitons were observed in a nonlinear
optical array by Eisenberg *at al* (1998). The experimental
results involving waveguide arrays are remarkable in how clearly
solitons are formed. Fig. 3 is a schematic of the waveguide array
and indicates that the input is a laser beam strongly focused
towards the center ridge. Fig. 4 shows the power measured for three
experiments at the end of the waveguide array. The experiments
correspond to low input power (70W), medium input power (320W) and
large input power (500W). One can see that at low power the beam diffracts
whereas focusing occurs at high input power. The bottom most portion
of the figure indicates strong self focusing and a discrete soliton
is formed.

Considerable interest involving the DNLS equation has appeared recently, following the experimental progress in the fields of nonlinear optical waveguide arrays and Bose-Einstein condensates trapped in periodic potentials arising from optical standing waves.

While DNLS equations are not transformable to the integrable discrete systems presented above, the latter nevertheless provide useful insight into discrete equations and solitary wave phenomena. For example, the use of soliton perturbation theory to elucidate the role of the on-site nonlinearity as a non-integrable perturbation to the IDNLS equation.

One can also consider discretizations of vector NLS equations, both integrable and nonintegrable. The integrable system is given by the following system \[\tag{11} i\frac{d\mathbf{q}_{n}}{dt}=\frac1{h^2}\left[ \mathbf{q}_{n+1} -2\mathbf{q}_{n}+\mathbf{q}_{n-1}\right] \pm\left| \left| \mathbf{q}_{n}\right| \right|^2\left( \mathbf{q}_{n+1} +\mathbf{q}_{n-1}\right) \]

where \(\mathbf{q}_{n}\) is an \(N\)-component vector. Eq.
(11) for \(\mathbf{q}_{n}=\mathbf{q}(nh)\) in the
limit \(h\rightarrow0,nh=x\) gives VNLS. The discrete VNLS system is
also integrable (Ablowitz *et al* 1999, Tsuchida *et al* 1999). The interested reader can find further details in
Ablowitz *et al* (2004a).

It is also worth mentioning that Eilbeck *et al* (1984) carried out an extensive study
(using path-following) of a coupled DNLS system (i.e., the vector generalization
of Eq.(10)) arising in crystalline acetanilide.

### Scalar discrete (2+1)-dimensional systems

In the \(2+1\)-dimensional case and for a square lattice, the DNLS equation is readily generalized to \[\tag{12} i\frac{d\psi_{n,m}}{dt}+\left| \psi_{n,m}\right| ^2\psi_{n,m}+\psi_{n+1,m} +\psi_{n-1,m}+\psi_{n,m+1}+\psi_{n,m-1}-4\psi_{n,m}=0 \qquad n,m\in \mathbb{Z} \]

and stationary solutions with frequency \(\omega\) can be found.
Surveys of recent results on the subject can be found in Kevrekidis
*et al* (2001) and Eilbeck and Johansson (2007).
Single-site peaked discrete soliton solutions of (12)
were first studied by Mezentsev *et al* (1994). The solution
can be smoothly continued from a single-site solution at the
anti-continuum (large-amplitude) limit \(\omega\rightarrow\infty\) to
the so called ground state solution of the continuous 2D NLS
equation in the small-amplitude limit \(\omega\rightarrow0\ .\) There is
an instability-threshold at \(\omega\sim1\ ,\) so that the solution is
stable for larger \(\omega\) (discrete branch) and unstable for
smaller \(\omega\) (continuous branch). The stability change is
characterized by a change of slope in the dependence of \(N(\omega)\ ,\)
where \(N(\omega)\) is the \(L^2\) norm of the mode
with frequency \(\omega\ .\)
The value of the excitation number \(N\)
at the minimum is nonzero, and thus there is an excitation threshold
for its creation (see Weinstein (1999)), in contrast to the 1D case.
The effect of this excitation threshold in 2D was proposed by
Kalosakas *et al* (2002) to be experimentally observable in
terms of a delocalizing transition of Bose-Einstein condensates in
optical lattices. The dynamics resulting from the instability on the
unstable branch is, in the initial stage, similar to the collapse of
the unstable ground state solution of the continuous 2D NLS
equation, with increased localization and blow-up of the central
peak. In contrast to the continuum case, however, the peak amplitude
must remain finite, and the result is a highly localized `pulson'
state where the peak intensity oscillates between the central site
and its four nearest neighbors. This process is referred to as
`quasicollapse' by Laedke *et al* (1994).

Finally, it is worth mentioning that two-dimensional lattices admit a new type
of localized solutions, `vortex-breathers', with no counterpart in
1D. For the DNLS model, vortex-breathers for a square 2D lattice
were first obtained by Johansson *et al* 1998. Typically they
become unstable as the coupling is increased.

## Dispersion management

As mentioned above, the NLS equation is an asymptotic approximation (via a quasi-monchromatic wave expansion) of Maxwell's equations with cubic nonlinear polarization terms. Recall that, for the NLS equation in the normalized form given by Eq. (3) above, \(q\) is related to the slowly varying complex envelope of the electromagnetic field in Maxwell's equations. Based on this relationship, it was predicted by Hasegawa and Tappert in 1973 that solitons could propagate in optical fibers; for additional references and historical background see Hasegawa and Kodama (1995) and Mollenauer and Gordon (2006).

Soliton propagation in fibers was demonstrated experimentally in 1980
in a seminal paper by Mollenauer, Stolen and Gordon, and work on
optical solitons continued in the following years.
Then, in a major development at the end of the 1980's, Erbium-doped
fiber amplifiers (EDFAs) were developed,
and started to be used in communication systems to counteract fiber loss.
The use of all-optical amplification eliminated the need for the electronic
regeneration of optical signals at various intervals throughout the system,
At the same time, however, it resulted in an optically transparent
transmission line, which meant that perturbations could now grow
unimpeded from the beginning to the end of the line.
Moreover, signal amplification by stimulated emission in always
accompanied by spontaneous emission which manifests as noise, and which
severely limited the transmission distances through which signals could
propagate successfully.
The use of frequency filters was proposed to reduce this difficulty.
In 1990's, system developers also started envisioning multi-channel
communication systems, in which signals in numerous carrier frequencies
are simultaneusly launched in the same fiber, usually in evenly spaced
frequency "windows", a technique called wavelength-division multiplexing (WDM).
The interactions between localized pulses in different channels, however,
resulted serious penalties associated with WDM transmission.
The technology of dispersion management was then developed
in order to deal effectively with these difficulties.
A dispersion-managed (DM) system consists in a periodic concatenation
of fibers with different dispersion characteristics, and turns
the large variations in the fiber dispersion to an advantage.
For example, in a two-step DM system, between every two optical amplifiers
(whose spacing is usually between 40 and 100km) one has a fiber with one
dispersion sign that is connected to another fiber with the opposite
dispersion sign.
Dispersion management is now widely used in modern communication systems,
and it significantly reduces the various penalties that arise in
multi-channel communications systems, such as four-wave mixing
(cf. Ablowitz *et al* (2003a)) and frequency and timing shifts
(cf. Ablowitz *et al* (2003b)). Moreover, dispersion management
also alleviates noise-induced penalties such as the Gordon-Haus and
Gordon-Mollenauer effects.
As a result, most long-distance optical fiber submarine systems today
are dispersion-managed.
A schematic diagram of a DM system is given in Fig. 5.
The figure shows the multiplexed input, a short precompensation fiber,
amplifiers (EDFA's: erbium doped fiber amplifiers), two fibers with
different dispersion characteristics between each successive EDFA,
and a short precompensation fiber before demultiplexing. The
pre- and post-compensation fiber sections are used to adjust for
the net average dispersion.

Mathematically speaking, the normalized equation used to describe DM systems is the NLS equation with rapidly varying coefficients, given by \[\tag{13} iu_z+\frac{d(z)}2u_{tt} +g(z)|u|^2u=0 \]

where \(d(z)\) is the dispersion, which in communications applications is large, and \(g(z)\) is the loss-gain coefficient. For example, in a two-step DM system, between every two optical amplifiers (say 40km apart) has one part of the fiber with one dispersion sign,; i.e. \(d(z)=d_1\) which is fused to the remaining part which has the opposite dispersion sign \(d(z)=d_2\) with \(d_1d_2<0\ .\) The choice of normalizations can be made such that the loss-gain coefficient has unit average in a dispersion map. A good understanding of the dynamical behavior is usually obtained by studying the "lossless" case \(g(z)=1\ .\)

As mentioned above, NLS equations are derived by asymptotic analysis from Maxwell's equations. Given the disparate scales involved in Maxwell's equations, i.e. operable wavelengths are near 1.5\(\mu\)m and transmission distances up to 10,000 km yield differing scales on the order of \(10^{13}\ ;\) hence asymptotic analysis and reductions to NLS equations are critically important. Researchers do not -- cannot, in fact -- solve Maxwell's equations through the exceedingly long distances that are required when dealing with such transmission problems. Numerical studies of the NLS equation in these DM environments show that localized pulse solutions exist; they are termed DM solitons. Asymptotic analysis (via the method of multiple scales) of such NLS equations with rapidly varying coefficients leads one to a nonlocal equation which governs strongly dispersion-managed communications systems. This asymptotic equation, termed the DMNLS equation (dispersion managed nonlinear Schrodinger equation) is nonlocal and it admits solitary wave solutions. In Fig. 6 below we show both a classical (sech profile soliton of (3) and a DM soliton obtained from the DMNLS equation obtained from Eq. (13) (see Gabitov and Turitsyn (1996) and Ablowitz and Biondini (1998)).

DMNLS has recently been an active field of research, and the interested reader
can find details in Kodama (2000), Lushinkov (2001), Ablowitz *et al* (2002)
and references therein.

Interestingly DM systems also give rise to other special solutions
of the DMNLS equation, in particular quasi-linear pulse solutions
(see Ablowitz *et al* 2001a, 2002) which are more like linear
modes, but have a nonlinear phase contribution, and are used more
frequently in communication applications than DM solitons.
Dispersion management is also used to generate ultra-short soliton
pulses on the order of a few femto-seconds in mode-locked lasers
such as Ti-sapphire, Sr-fosterite and fiber lasers, amongst others.
Here the dispersion compensation is designed to allow the system to
be in the net anomalous dispersion regime. In Fig. 7 a typical
mode-locked Ti-sapphire laser system is depicted and in Fig. 8
an associated pulse train.

Typical values are the following. Pulse width\[\tau= 10\]fs\( =10^{-14}\)sec; repetition time between pulses\[T_{rep}=10\]ns\( =10^{-8}\) sec and repetition frequency\[f_{rep} =\frac1{T_{rep}}=100\] MHz (MHz: megahertz).

## Mathematical Framework

### Inverse Scattering Transform (IST)

As mentioned above, there are a number of equations of NLS type,
both continuous and discrete, which are solvable by IST.
More precisely, the following NLS-type equations are
solvable by IST: the scalar NLS equation in 1+1 dimensions
[i.e., Eq. (3)]
for both rapidly decaying initial data (\(q \rightarrow 0\)
as \(t\to\pm\infty\))
and for data which decays tends to a constant background
(\(|q|\to|q_0|\) as \(t\to\pm\infty\ ,\)
with \(q_0 \neq 0\)); the vector NLS equation in 1+1 dimensions
(6), also both for rapidly decaying initial data,
and for data which decays rapidly to a constant background);
the scalar and vector integrable discrete NLS equations
-- i.e., Eqs. (9) and (11)
for decaying data, and, in the scalar case, also for nondecaying data.
Finally, a special case of the multi-dimensional Benney-Roskes equation,
called the Davey-Stewartson equation
[namely, Eq. (8) with \(\sigma_1,\sigma_2=\pm1\)]
is also solvable by IST.
There are various references to the subject; cf. Ablowitz and Segur (1981),
Calogero and Degasperis (1982), Faddeev and Takhtajan (1987),
Novikov *et al* (1984), Ablowitz and Clarkson (1991),
Ablowitz *et al* (2004a).

The IST is a method that allows one to linearize a class on nonlinear evolution equations. An essential pre-requisite of the IST method is the association of the nonlinear PDE with a pair of linear problems. We say that the operator pair \(\mathbf{X}, \mathbf{T}\) is a Lax pair for the nonlinear equation \[ q_t=F[x,t,q,q_x,q_{xx},\dots]\, , \qquad q=q(x,t)\, , \] if the compatibility condition of the overdetermined linear system of differential equations \[ v_x=\mathbf{X} v\, , \qquad v_t=\mathbf{T} v \] [i.e., the equality of the mixed derivatives \(v_{xt}=v_{tx}\)] is identically satisfied provided \(q\) solves the nonlinear PDE. Here \(\mathbf{X},\mathbf{T}\) are in general matrix functions of \(q,q_x,q_{xx},\dots\) and of a complex parameter \(k\) which is assumed to be time-independent.

The solution of the initial-value problem by IST proceeds in three steps, as follows:

- the direct problem - the transformation of the initial data from the original "physical" variables (\(q(x,0)\)) to the transformed "scattering" variables (\(S(k,0)\));
- time dependence - the evolution of the transformed data often according to simple, explicitly solvable evolution equations (i.e., finding \(S(k,t)\));
- the inverse problem - the recovery of the evolved solution (\(q(x,t)\)) from the evolved solution in the transformed variables (\(S(k,t)\)).

Both the direct and the inverse problem make use of the first operator in the Lax pair, so-called scattering problem, while the time evolution is determined by the second operator in the Lax pair.

In the direct problem, the first step is to construct eigenfunction
solutions of the associated linear problem. These eigenfunctions
depend on both the original spatial variables and the spectral
parameter (eigenvalue) \(k\ .\) Second, with these eigenfunctions, one
determines *scattering data* that are independent of the
original spatial variables. In the inverse problem, the first step
is the recovery of the eigenfunctions from the (evolved) scattering
data. Finally, one recovers the solution in the original variables
from these (evolved) eigenfunctions.

The analyticity properties of the eigenfunctions in terms of the scattering parameter are key to the formulation of the inverse problem. For the NLS systems discussed above, the eigenfunctions are sectionally analytic functions of the scattering parameter and the inverse problem is therefore formulated as a generalized Riemann-Hilbert problem, which in turn is transformed into a system of linear algebraic-integral equations.

In conclusion, we mention that with the IST machinery one can do more than solve the initial-value problem; one can also construct special solutions by positing an elementary ansatz for the transformed variables and then applying the inverse transformation to obtain the corresponding solution in the physical variables. In general, the soliton and multisoliton solutions are constructed in this way. Also, the IST provides an effective way to study the asymptotic (long-term) behavior of the solutions (e.g., via the nonlinear steepest descent technique), as well as a way to study their stability (using the so-called squared eigenfunctions).

### Direct Methods

Solutions to certain nonlinear PDEs can be obtained by using direct methods. For instance, one can effectively transform the nonlinear equation into a bilinear differential equation, often called the Hirota bilinear form, and then find exact solutions by a perturbation technique. In the case of the NLS equation (Hirota 1973), substituting \(u= G/F\ ,\) \(F\) real, into \(iu_t+u_{xx}+|u|^2u=0\ ,\) gives the bilinear form \[ \frac{1}{F^2}(iD_t+D_x^2)G\cdot F-\frac{G}{F^3}(D_x^2F\cdot F-GG^*)=0 \] where the bilinear operator is defined by \[ D_x^m D_t^n a\cdot b=\left. (\partial_x -\partial_{x^{\prime}})^m (\partial_t -\partial_{t^{\prime}})^n a(x,t)b(x^{\prime},t^{\prime})\right|_{x^{\prime}=x, \ t^{\prime}=t}. \] The bilinear equation can be decoupled by choosing \[ (iD_t+D_x^2)G\cdot F=0, \qquad D_x^2F\cdot F=GG^*, \] whereupon \[ |u|^2=\frac{GG^*}{F^2}=\frac{D_x^2 F\cdot F}{F^2}=2(\log F)_{xx} \] A one soliton solution can then be obtained by taking \(G=e^{\eta_1}\) and \(F=1+e^{\eta_1+\eta_1^*+\phi_{1,1}}\ ,\) with \({\eta_1}=p_1x-\Omega_1t +x_0\ ,\) \(\Omega_1=-ip_1^2\) and \(e^{\phi_{1,1}}=(p_1+p_1^*)^{-2}/2\ .\) Multisoliton solutions can be obtained in a similar way.

Other direct methods, such as Backlund and Darboux transformations, can be applied to obtain explicit solutions of certain nonlinear PDE's (Matveev and Salle, 1991). Darboux transformations for the NLS equation have been obtained by Mañas (1996).

For an algebro-geometric approach to the theory of integrable equations we refer
the interested reader to the monographs by Belokolos *et al* (1994) and Gesztesy and Holden
(2003).

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