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Nonlinear Schrodinger systems: continuous and discrete

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   Mark Ablowitz and Barbara Prinari (2008), Scholarpedia, 3(8):5561.
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   Curator: [7]Mark Ablowitz
   Contributors:


   0.62 -

   [8]Eugene M. Izhikevich
   0.25 -

   [9]Barbara Prinari
   0.12 -

   [10]Tobias Denninger
   0.12 -

   [11]Nick Orbeck

   [12]Gino Biondini

   [13]Norman J. Zabusky

   [14]Chris Eilbeck
     * [15]Dr. Mark Ablowitz, Department of Applied Mathematics, University of
       Colorado, Boulder, CO
     * [16]Barbara Prinari, Dipartimento di Fisica and Sezione INFN Universita del
       Salento

   The Nonlinear Schrodinger (NLS) equation is a prototypical dispersive nonlinear
   [17]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 [18]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 [19]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 [20]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 [21]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
   [22]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 [23]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.

Contents

     * [24]1 Continuous models
          + [25]1.1 Scalar (1+1)-dimensional systems
          + [26]1.2 Vector (1+1)-dimensional systems
          + [27]1.3 Scalar multidimensional systems
          + [28]1.4 Benney-Roskes and Davey-Stewartson models
          + [29]1.5 Vector multidimensional systems
     * [30]2 Discrete models
          + [31]2.1 Discrete (1+1)-dimensional systems
          + [32]2.2 Scalar discrete (2+1)-dimensional systems
     * [33]3 Dispersion management
     * [34]4 Mathematical Framework
          + [35]4.1 Inverse Scattering Transform (IST)
          + [36]4.2 Direct Methods
     * [37]5 References
     * [38]6 See also

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
   [39]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 \[\tag{1}
   k-k_0=k'(\omega_0)(\omega-\omega_0)+\frac{k''(\omega_0)}2(\omega-\omega_0)^2+\fra
   c{\omega_0n_2}{c}|E|^2 \]

   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. ([40]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.([41]3), and
   it corresponds to anomalous dispersion. The defocusing case obtains when the
   dispersion is normal, and it corresponds to the (-) sign in Eq.([42]3).

   The NLS equation possesses [43]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 \(\eta=1/\sqrt2\)).
   Figure 1: A bright soliton\[A^2\] is the square modulus of the solution,
   \(\zeta\) is the coordinate in the moving frame.

   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 \(\alpha=
   \pi/3\)).
   Figure 2: A gray soliton\[A^2\] is the square modulus of the solution, \(\zeta\)
   is the coordinate in the moving frame.

   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
   [44]averaging over the fast birefringence [45]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 ([46]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. ([47]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.([48]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 [49]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 (\(\sigma d0\ ,\) whereas in water waves \(\rho0\) 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
   [51]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.
   Figure 3: Optical waveguide structure.
   Figure 4: Output power at the end of the waveguide array corresponding to
   different input power values.

   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
   [52]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. ([53]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.([54]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 ([55]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
   [56]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. ([57]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
   1980s, 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 simultaneously 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.
   Figure 5: Schematic diagram of a dispersion-managed communication system.

   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