Nonlinear Schrodinger systems: continuous and discrete
| Mark Ablowitz and Barbara Prinari (2008), Scholarpedia, 3(8):5561. | doi:10.4249/scholarpedia.5561 | revision #137230 [link to/cite this article] |
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.
Contents |
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 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.
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).
References
- Ablowitz MJ, Bakirtas I, Ilan B (2005), Physica D 207 230
- Ablowitz MJ and Biondini G (1998), Optics Letters 23 1668
- Ablowitz MJ, Biondini G, Blair S (1997), Phys Lett A 236 520
- Ablowitz MJ, Biondini G, Blair S (2001), Phys Rev. E 63 046605
- Ablowitz MJ, Biondini G, Chakravarty S, Horne RL (2003a), J. Opt. Soc. Am. B 20 831
- Ablowitz MJ, Biondini G, Chakravarty S, Jenkins RB (1996), Optics Letters 21 1646
- Ablowitz MJ and Clarkson PA (1991), Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math Soc Lecture Notes Series vol 149, Cambridge Univ Press
- Ablowitz MJ, Docherty A and Hirooka T (2003b), Optics Letters 28 1191
- Ablowitz MJ and Hirooka T (2002a), JOSAB 19 425
- Ablowitz MJ, Hirooka T and Biondini G (2001b), Optics Letters 26 459
- Ablowitz MJ, Hirooka T and Inoue T (2002b) J. Opt. Soc. Am. B 19, 2876
- Ablowitz MJ and Ladik JF (1975), J Math Phys 16 598
- Ablowitz MJ and Ladik JF (1976), J Math Phys 17 1011
- Ablowitz MJ, Musslimani Z (2003c), Phys. Rev. E 67 025601(R)
- Ablowitz MJ, Ohta Y, Trubatch AD (1999), Phys Lett A 253 287
- Ablowitz MJ, Prinari B, Trubatch AD (2004a), Continuous and Discrete Nonlinear Schrodinger systems, London Math Soc Lecture Notes Series vol 302, Cambridge Univ Press
- Ablowitz MJ, Prinari B, Trubatch AD (2004b), Inv Probl 20 1217
- Ablowitz MJ, Segur H (1981) Solitons and the Inverse Scattering Transform, SIAM 4
- Ablowitz MJ, Segur H (1979), J Fluid Mech 92 691
- Agrawal GP (2001) Nonlinear Fiber Optics, Third Edition. Academic Press, New York
- Anastassiou C, Segev M, Steiglitz K, Giordmaine MA, Mitchell M, Shih M, Lan S, Martin J (1999), Phys. Rev. Lett. 83 2332
- Belokolos ED, Bobenko AI, Enolski VZ, Its AR, Matveev VB (1994), Algebro-geometric approach in the theory of integrable equations, Springer, Berlin
- Benney DJ, Newell AC (1967), J Math Phys 46 133
- Benney DJ, Roskes GJ (1969), Studies App Math 48 377
- Calogero F, Degasperis A (1982) Spectral Transform and Solitons. I Amsterdam: North-Holland
- Chen M, Tsankov MA, Nash JM and Patton CE (1994), Phys. Rev. A 49 12773
- Chiao RY, Garmire E, Towner CH (1964), Phys Rev Lett 15 479
- Christodoulides DN and Joseph RI (1988), Optics Lett 13 794
- Davey A, Stewartson K (1974), Proc Royal Soc Lon Series A 338 101
- Davydov AS (1973), J Theor Biol 38 559
- Efremidis NK, Christodoulides DN, Fleischer JW, Cohen O, Segev M (2003), Phys. Rev. Lett. 91 213906
- Eilbeck JC, Lomdhal PS, Scott AC (1984), Phys Rev B 30, 4703
- Eilbeck JC, Lomdhal PS, Scott AC (1985), Physica D 16, 318
- Eilbeck JC, Johansson M (2003), Proceedings of the Third Conference ``Localization and Energy Transfer in Nonlinear Systems, eds L. Vazquez, R. S. MacKay, M. P. Zorzano, World Scientific, Singapore, 44--67, 2003', also arXiv:nlin/0211049v1
- Eisenberg HS, Silberberg Y, Morandotti R, Boyd AR, and Aitchison JS (1998), Phys Rev Lett 81 3383
- Faddeev LD, Takhtajan LA (1987) Hamiltonian Methods in the Theory of Solitons Springer Verlag
- Gabitov IR, Turitsyn SK (1996), Opt Lett 21, 327
- Gesztesy F and Holden H (2003), Soliton Equations and Their Algebro-Geometric Solutions 79 Cambridge Univ Press
- Ginibre J and Velo G (1979), J Func Anal 32, 1
- Ginzburg VL (1956), Sov Phys JETP 2 589
- Ginzburg VL, Landau LD (1950), Sov Phys JETP 20 1064
- Ginzburg VL, Pitaevskii LP (1958), Sov Phys JETP 7 858
- Gordon JP, Haus HA (1986) Opt Lett 11 665
- Gordon JP, Mollenauer LF (1990) Opt Lett 15 1351
- Hasegawa A, Tappert F (1973a), App Phys Lett 23 142
- Hasegawa A, Tappert F (1973b), App Phys Lett 23 171
- Hasegawa A, Kodama Y (1995), Solitons in Optical Communciations Oxford University Press, Oxford.
- Hirota R (1973), J Math Phys 14, 805
- Johansson M, Aubry S, Gaididei YuB, Christiansen PL, Rasmussen K{\O} (1998), Physica D 119 115
- Kalosakas G, Rasmussen K{\O}, Bishop AR (2002), Phys Rev Lett 89 030402
- Kelley PL (1965), Phys Rev Lett 15 1005
- Kevrekidis PG, Rasmussen KA, Bishop AR (2001), Int J Mod Phys B 15 2833
- Kivshar YS, Turitsyn SK (1993), Opt Lett 18 337
- Kodama Y (2000), in Massive WDM and TDM Soliton Transmission Systems Hasegawa Ed. (Kluwer Academic) p. 129
- Laedke EW, Spatschek KH, Turitsyn SK (1994), Phys Rev Lett 73 1055
- Lushnikov PM (2001), Opt. Lett. 26, 1535
- Manakov SV (1974), Sov Phys JETP 38 248
- Mañas M (1996), J Phys A: Math Gen 29, 7721
- Matveev VB, Salle MA (1991), Darboux transformations and solitons, Springer, Berlin
- Menyuk CR (1987), IEEE J Quant Elect 23 174
- Menyuk CR (1999), J Eng Math. 36 113
- Merle F and Raphael P (2004), Inventiones Mathematicae 156, 165
- Mezentsev VK, Musher SL, Ryzhenkova IV, Turitsyn SK (1994), JETP Lett 60 829
- Molleneauer LF and Gordon JP (2006) Solitons in Optical Fibers: Fundamental and Applications to Telecommunications Academic Press, London
- Molleneauer LF, Stolen RH, Gordon JP (1980) Phys Rev Lett 45 1095
- Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE (1984) Theory of Solitons. The Inverse Scattering Method New York: Plenum
- Papanicolaou GC, Sulem C, Sulem PL, Wang XP (1994), Physica D 72, 61
- Pethick CJ and Smith H (2002) Bose- Einstein Condensation in Dilute Gases, Cambridge University Press
- Prinari B, Ablowitz MJ, Biondini G (2006), J Math Phys 47 063508
- Roskes GJ (1976), Stud App Math 55 231
- Soljacic M, Steiglitz K, Sears SM, Segev M, Jakubowski MH, Squier R (2003), Phys Rev Lett 90 254102
- Su WP, Schieffer JR and Heeger AJ (1979), Phys Rev Lett 42 698
- Sulem C, Sulem PL (1999) The nonlinear Schrodinger equation - Self-focusing and Wave Collapse, Applied Mathematical Sciences vol 139 Springer
- Talanov VI (1964), Radiophysics 7 254
- Talanov VI (1965), Sov Phys JETP Lett 109 138
- Tsuchida T, Ujino H, Wadati M (1999), J Phys A 32 2239
- Vlasov F, Petrishchev V, Talanov V (1971), Radiophys. Quant. Elec. 14 1070
- Weinstein MI (1999), Nonlinearity 12 673
- Zakharov VE (1968), Sov Phys J App Mech Tech Phys 4 190
- Zakharov VE (1972), Sov Phys JETP 35 908
- Zakharov VE, Shabat AB (1972), Sov Phys JETP 34 62
- Zakharov VE, Shabat AB (1973), Sov Phys JETP 37 823
- Zvedzin AK, Popkov AF (1983), Sov Phys JETP 57 350
Internal references
- Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
- Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
