Chapter 1: Introduction

Published:2022
Boris A. Malomed, "Introduction", Multidimensional Solitons, Boris A. Malomed
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This chapter offers an introduction to the vast area of experimental and theoretical studies of solitons. The chapter is composed of two large sections. The first one provides a review of effectively 1D settings, before proceeding to solitons in two and threedimensional spaces. The body of theoretical and experimental results accumulated for 1D solitons is really large, with the most essential among them overviewed in the first section. For this reason, it is quite long. The exit to the realm of multidimensional solitons is offered in the second section of the chapter. Both sections are split into a number of subsections, which clearly define particular settings and problems addressed by them. In addition to its role as the gateway to the whole book, this chapter may be used by those who are interested in a reasonably short, but, nevertheless, sufficiently detailed introduction to the modern “soliton science.”
1.1 OneDimensional Solitons
1.1.1 The dawn of the soliton era: The Korteweg–de Vries (KdV) equation
The word “soliton,” derived from “solitary wave,” is a common name for an extremely broad class of localized objects existing in a great variety of physical media, which feature interplay of basic linear properties, such as dispersion and/or diffraction, and nonlinearity, which gives rise to selfattraction of matter or fields that fill the media. Simultaneously with the progress of experimental research of solitons in an impressive number of physical realizations, a great deal of work has been done on theoretical models of solitons (actually, as it usually happens, the progress in the theoretical work was much faster). The theory has been developing in two closely related but distinct streams: on one hand, elaboration of mathematical models of diverse physical setups, to which the concept of solitons is relevant, and, on the other hand, mathematical investigation of these and many other models, many of which have been introduced on the basis of the mathematical interest, rather than being suggested by experimental motivations.
The first report of the observation of what is nowadays called a soliton was published by Scott Russell (1844), the British (Scottish) civil engineer and naval architect. Traveling on horseback along the Union Canal in Edinburgh in 1834, he observed accidental generation of a “heap of water” (solitary wave) by a barge that was traveling along the canal and suddenly came to a halt. According to the report, the solitary wave had a length of ca. 9 m, a height of ca. 40 cm, and it had traveled the distance ≃2 km before disappearing (Scott Russell followed it on horseback; in his report, the estimates were given in imperial units). He understood the significance of the observation of what he named “a wave of translation” and built a water tank at his home for running experiments with such waves. In the experiments, he had identified their basic properties, such as remarkable stability, dependence of the speed on the height, the ability to pass through each other, and splitting of an input, which was “too big” for generating a single solitary wave, in two separate waves traveling at different speeds. The famous observation of John Scott Russell was reproduced, with the help of a motorboat, in the same canal in 1995, as a part of a conference on solitons at the HeriotWatt University (see Fig. 1.1).
Originally, the report of Scott Russell was considered skeptically. However, when the famous Korteweg–de Vries (KdV) equation was published by its authors in 1895, as a fundamental model for the propagation of long smallamplitude waves on the surface of shallow inviscid water, it became clear that the “translation wave,” discovered by Scott Russell, exactly corresponded to the elementary solitarywave solution of the KdV equation [in fact, the same equation was published about 20 years earlier by Boussinesq (1877), but, being hidden in the depth of a huge “memoir,” his derivation was discovered much later].
The standard normalized form of the KdV equation for local deformation −u(x, t) of the surface of the water layer (it is a real variable) is
where t and x are, as usual, the scaled evolutional (temporal) and spatial (coordinate) variables, and the subscripts stand for partial derivatives. The original paper by Korteweg and de Vries (1895) reported a family of elementary solitarywave solutions of Eq. (1.1), parameterized by arbitrary velocity c > 0 (the KdV equation admits only unidirectional propagation),
[recall the definition of the hyperbolic secant: sech z ≡ 1/cosh z ≡ 2/(e^{z} + e^{−z})], as well as periodic solutions written in terms of Jacobi’s elliptic function cn (for this reason, such solutions are often called “cnoidal waves”). In fact, solutions (1.2) with all values of c > 0 are tantamount to each other, because Eq. (1.1) is invariant with respect to the scaling transform,
where x_{0} is an arbitrary scaling factor. This transformation with $x0=c2/c1$ relates solitons (1.2) with positive velocities c_{1} and c_{2}.
An important fact is that KdV soliton (1.2) is completely stable against perturbations. A spatiotemporal plot of the soliton with c = 4 is shown in Fig. 1.2(a).
The KdV equation is actually a universal dynamical model (“normal form”) for the propagation of weakly nonlinear long waves in conservative media. Accordingly, in addition to the waves on shallow water, the same equation finds other fundamentally important realizations—in particular, in the application to internal waves in stratified fluids (Sutherland, 2013) and ionacoustic waves in plasmas (Swanson, 2003) [see also the book by Dauxois and Peyrard (2006)].
A great deal of interest was drawn to the KdV equation 70 years since its publication, by the work of Zabusky and Kruskal (1965), who simulated collisions of solitons (1.2) with different velocities and discovered that, unexpectedly, the solitons demonstrated absolutely elastic collisions, passing through each other in an unscathed form. In that paper, the word “soliton” was coined, and this numerical observation was a key incentive for the discovery, two years later, by Gardner et al. (1967), of the mathematical miracle in the form of the exact integrability of the KdV equation by means of the inversescattering transform (IST). This method provides an extremely powerful tool for constructing an infinite variety of exact solutions of the KdV equation, including those accounting for elastic collisions between two or arbitrary number of solitons. This discovery had also offered an explanation to what was considered as another miracle, viz., apparent lack of chaotization in simulations of the nonlinear Fermi–Pasta–Ulam (Fermi et al., 1995) lattice which is, as a matter of fact, a discrete version of the KdV equation (not a truly integrable one, but close to the integrability).
As an illustration of highly nontrivial exact solutions of the KdV equation produced by the IST, one can take a solution generated by input
with integer $N$. It generates a complex but exact analytical solution, which may be construed as a nonlinear superposition of $N$ elementary solitons (1.2) with different velocities. In particular, for $N=2$ the twosoliton solution takes a relatively simple form, which is relevant for t > 0 and t < 0 alike,
In fact, as one can see in the plot of this solution, displayed in Fig. 1.2(b), it represents the collision of two solitons with velocities c_{1} = 16 and c_{2} = 4, with the maximum overlap of the colliding pulses taking place at t = 0.
More general inputs, such as the one given by Eq. (1.4) with coefficients different from the specific value $N(N+1)$, generate a mixture of solitons and quasilinear waves, alias “radiation modes,” which spread in the course of the evolution. In terms of the IST for KdV and other integrable equations, solitons are represented by the discrete spectrum of the direct scattering problem, while the radiation component corresponds to the continuous spectrum.
Another exact result for the KdV equation is that, while the collision between two solitons with velocities c_{1} and c_{2} (assuming that, originally, the faster one with a larger velocity, c_{1}, is placed behind the slower counterpart), does not change the shapes or velocities of the solitons, it produces their finite shifts: the faster one jumps forward by the distance
while the slower soliton bounces backward by
It is relevant to compare these results with similar ones for collisions between sineGordon kinks (which are also produced by the solution of an integrable equation) [see Eq. (1.79) below].
As a consequence of the exact integrability, the KdV equation sustains an infinite set of dynamical invariants (conserved quantities). Three lowest ones have direct physical interpretation: the total norm N (in terms of water waves, it measures the mass carried by them) and momentum P,
and the Hamiltonian,
Note that the KdV equation can be written in terms of the Hamiltonian in the following form:
where δ/δu stands for the variational (Fréchet) derivative [for the mathematical presentation of the variational derivative, see the book by Cassel (2013)]. As concerns the surface waves modeled by the KdV equation, both the local deformation of the surface, i.e., the mass density of the flow involved in the wave, and the local velocity of the longitudinal flow are proportional to variable u(x, t), as shown in the classical work by Korteweg and de Vries (1895). This fact explains why the momentum density in Eq. (1.6) is proportional to u · u ≡ u^{2}.
Starting from the pioneering work of Gardner et al. (1967), the IST method has been a subject of an ever growing number of mathematical works, leading to the discovery of many other integrable equations and many species of highly nontrivial exact solutions. Techniques and results produced by these works are summarized in wellknown books by Zakharov et al. (1980), Ablowitz and Segur (1981), Calogero and Degasperis (1982), Newell (1985), Takhtadjian and Faddeev (1986), Yang (2010), and others.
1.1.2 The next stage of the soliton history: The nonlinear Schrödinger (NLS) equation
1.1.2.1 The mathematical basis
The next fundamentally important step in the development of the IST method and identification of equations to which it applies was its elaboration for the nonlinear Schrödinger (NLS) equations with the cubic term, reported in the classical work by Zakharov and Shabat (1971). Another fundamentally important contribution to the development of the IST technique was made by Ablowitz, Kaup, Newell, and Segur (1974; known as AKNS). The normalized form of the NLS equations for complex wave functions ψ(x, t) is
where signs + and − represent, respectively, the selfattractive and selfrepulsive (alias focusing and defocusing) nonlinearity. Equation (1.3) with the selfattraction term gives rise to the commonly known family of fundamental brightsoliton solutions (“bright” implies that, in terms of the realization in optics, the soliton seems as a pulse or beam of light against the dark background),
where real constant η is an arbitrary amplitude of the soliton. The same constant determines the soliton’s width, the usual definition for which is FWHM (half width at halfmaximum),
where x_{max} is the point at which the local intensity of the wave field, ψ(x, t)^{2}, has its maximum. Obviously, for soliton (1.4), this definition gives
Real parameter c in solution (1.10) represents an arbitrary velocity of the NLS soliton [unlike the KdV soliton (1.2), the velocity may be both positive and negative]. In fact, the presence of this parameter is a manifestation of a more general property of the Galilean invariance of the NLS equation: if ψ(x, t) is any solution, a family of boosted ones is generated by the Galilean transform,
Further, similar to what is said above about the KdV solitons, all NLS ones are tantamount to each other, due to the invariance of Eq. (1.9) with respect to the scaling transformation [cf. its counterpart (1.3) for the KdV equation],
The profile of soliton (1.4) with η = 1 and c = 0 is displayed in Fig. 1.3(a). Collisions between the NLS solitons with different velocities give rise to shifts of their positions and phases. In particular, solitons with equal amplitudes η, colliding with velocities ±c, shift in the direction of their motion by Δx = ±η^{−1} ln(1 + 4η^{2}/v^{2}).
Similar to the KdV equation, which admits exact multisoliton solutions generated by the specially chosen input (1.4), it was demonstrated by Satsuma and Yajima (1974) that NLS equation (1.9) with the top (selffocusing) sign gives rise to highly nontrivial exact solutions in the form of $N$solitons, produced by the initial condition,
with integer $N$. These states may be considered as nonlinear superpositions of $N$ fundamental solitons (1.10), with the set of amplitudes,
each with c = 0. A difference from the exact multisoliton solution of the KdV equation [see Eq. (1.5)] is that individual constituents building the $N$soliton of the NLS equation do not separate. They stay together, forming an oscillatory state, which is usually called a breather. This solution can be written in a relatively simple form for $N=2$,
As shown in Fig. 1.3(b), the twosoliton breather periodically oscillates between the broad and narrow shapes, the latter one featuring a single central peak and two small side peaks. For $N=3$ in Eq. (1.15), the analytical form of the resulting threesoliton is cumbersome, while its spatiotemporal evolution, displayed in Fig. 1.4(a), along with the respective Fourier transform in Fig. 1.4(b), exhibit a new feature, in comparison with the twosoliton: the central peak periodically splits in two, which then recombine back.
NLS equation (1.9) with the bottom sign, which represents the selfdefocusing cubic nonlinearity, does not have brightsoliton solutions, but it gives rise to dark solitons. In terms of optics, they represent a dark spot on top of the uniformly lit backdrop,
where n_{0} is the intensity of the extended background supporting the dark soliton, which moves across the background with positive or negative speed c, subject to constraint c^{2} < n_{0}.
Similar to KdV, the integrable NLS equation (1.9) conserves an infinite set of dynamical invariants, the three lowest ones having a straightforward physical interpretation: the integral norm and momentum [cf. Eq. (1.6) for KdV],
with * standing for the complex conjugate, and the Hamiltonian [cf. Eq. (1.7)],
Note that, for the field configurations vanishing at infinity, i.e., ones satisfying the condition
the application of the integration by parts to expression (1.18) for the momentum shows that its value if real, P* = P, even if the expression seems complex. Further, the NLS equation (1.9) can be written in terms of their Hamiltonian [cf. the Hamiltonian representation of the KdV equation given by Eq. (1.8)],
where ψ and ψ* are considered as independent arguments of the integral functional H given by Eq. (1.19).
The total norm and Hamiltonian of input (1.15) with integer $N$ are
Further, it is easy to check that the total norm and Hamiltonian of the set (1.15′) are exactly equal to values (1.19′). This fact implies that, although the breather generated by input (1.15) may be considered as a bound state of the set of $N$ fundamental solitons (1.15′), the binding energy of this complex is exactly equal to zero; hence, it is expected to be fragile. Indeed, small perturbations can easily split the breather into a set of separating fundamental solitons, as shown in detail by Marchukov et al. (2019, 2020).
Both the bright and dark solitons are stable solutions of the respective NLS equations. Moreover, the bright soliton realizes the ground state of the setup, i.e., the state with lowest value of the Hamiltonian for a fixed norm (Zakharov and Kuznetsov, 2012).
The input in the form given by Eq. (1.15) is relevant to the experiment in the general case, when $N$ is not an integer. In this case, an explicit solution for ψ(x, t) is not available, but the respective set of scattering data, in terms of the IST method, was found in an exact form by Satsuma and Yajima (1974). The set contains a multiple soliton (breather) of order $Nsol=[N+1/2]$ ([ · · · ] stands for the integer part), “contaminated” by the dispersive radiation component. Thus, input (1.15) with
creates the output containing exactly one soliton.
In the absence of an explicit analytical solution for this case, the variational approximation (VA) can be used, as proposed by Anderson (1983) and Anderson et al. (1988) (see also a review by Malomed, 2002). The VA is based on the Lagrangian from which Eq. (1.9) (with the top sign in front of the cubic term) can be derived,
The solution is approximated by the Ansatz in the form of a sech solitonlike pulse [cf. Eq. (1.10)], with an arbitrary relation between the real amplitude A and width a, both of which may be functions of time. In addition, the phase structure of the Ansatz includes a real timedependent chirp, b(t),
The substitution of the Ansatz in Lagrangian (1.23) yields the VA Lagrangian,
The Euler–Lagrange equations produced by Lagrangian (1.25) as per the standard rules of the variational calculus (Cassel, 2013) confirm the conservation of norm (1.18) of the Ansatz, (d/dt)(N_{sol} ≡ 2A^{2}a) = 0, and produce an expression for the chirp,
which demonstrates that the chirp is generated by the temporal dependence of the width. Finally, the evolution equation for the width takes the form of the equation of motion of the Newtonian particle of unit mass in an effective potential which is shown in Fig. 1.5(a),
This potential is tantamount to one in the classical Kepler’s problem for the radial coordinate of a planet moving under the action of the Sun’s gravity. The minimum of the potential, at a = 1/N_{sol}, exactly corresponds to the fundamentalsoliton solution (1.10). On the other hand, Eq. (1.27) predicts that, for given norm N_{sol}, the input in the form of Ansatz (1.24) with b = 0 produces the soliton under condition a > a_{0} ≡ 1/N_{sol}. On the other hand, the exact solution of Satsuma and Yajima (1974) produces a single soliton in interval (1.22); hence, the exact solution yields a milder condition for the formation of the soliton, viz., a > 1/(2N_{sol}).
1.1.2.2 Physical realizations of NLS equations: Plasmas and hydrodynamics
Relatively old physical realizations of the NLS equation are provided by Langmuir waves in hot plasmas [spatially modulated oscillations of free electrons relative to heavy ions (Swanson, 2003) and surface waves on deep water (Zakharov, 1968)]. These wave modes are drastically different from, respectively, ionacoustic waves in plasmas and surface waves on shallow water, which are modeled, as mentioned above, by the KdV equation. As concerns the Langmuir waves, it is relevant to mention that the NLS equation with the selffocusing sign of the cubic term is an adequate model for them in the adiabatic approximation, in which the ion density, n(x, t), directly follows the variation of the density, ψ(x, t)^{2}, corresponding to the electron wave function, ψ(x, t). A more accurate model is provided by the celebrated Zakharov’s system, which, unlike the NLS equation, is not integrable. Its normalized form is
The adiabatic approximation neglects term n_{tt} in Eq. (1.29), yielding n ≈ −ψ^{2}, the substitution of which in Eq. (1.28) reproduces the NLS equation (1.9) with the top sign.
1.1.2.3 Realizations of NLS and similar equations in optics
1.1.2.3.1 Cavities, fibers, and planar waveguides
NLS equations have very important realizations in optics [see books by Akhmediev and Ankiewicz (1997), Kivshar and Agrawal (2003), and Agrawal (2013)]. Precisely in the form of Eq. (1.9), with t and x being the time and coordinate, the NLS equation governs the evolution of the amplitude of the electromagnetic field in laser cavities, although a realistic model of the cavity includes losses and pump. If the latter factors are taken into regard, the NLS equation is replaced by one introduced by Lugiato and Lefever (1987),
where term ψ_{xx} represents paraxial diffraction, α > 0 is the loss constant, E is the pump field (in the simplest case, it is a given constant), and Δ, which may be positive or negative, is the pump’s detuning with respect to the cavity’s eigenfrequency.
In this connection, it is relevant to mention that the generation of dissipative optical solitons in cavities with the Kerr nonlinearity (often called microresonators), which gives rise to spectra in the form of phaselocked frequency combs, has drawn a great deal of interest (Kippenberg et al., 2018; Kues et al., 2019; Diddams et al., 2020; and Wang et al., 2020). The combs constitute the basis of several advanced technologies in integrated photonics, such as precision spectroscopy, LIDAR (laser imaging, detection, and ranging), and optical data processing. The basic model for the dynamics of solitons in dissipative Kerr microresonators is similar to Eq. (1.30), with x being a circular coordinate in the microresonator. The equation may include additional spatial derivatives, accounting for higherorder spatial dispersion.
As concerns the topic of 3D spatiotemporal solitons [often called “light bullets” (LBs), following Silberberg (1990)], stable 3D solutions of Eq. (1.30) of this type, as well as clusters of such LBs, were reported by Gopalakrishnan et al. (2021). The number of LBs created by Eq. (1.30) depends on initial conditions, while the LB’s peak power is determined by the parameters of the equation. These LBs are stable provided that the pump is not too strong. In the framework of semidiscrete 2D and 3D versions of the Lugiato–Lefever equation (1.30), with one continuous and one or two discrete coordinates, stable semidiscrete LBs were found, in a numerical form, by Panajotov et al. (2021).
Another famous realization of the NLS equation in optics was predicted by Hasegawa and Tappert (1973) in the form of the propagation equation for the envelope of electromagnetic waves in nonlinear optical fibers. In that case, temporal variable t in the NLS equation is replaced by the propagation distance, z, while the spatial coordinate x is replaced by its temporal counterpart (local time),
where V_{gr} is the group velocity of the carrier wave. In silica fibers, the nonlinearity is always selffocusing, induced by the material Kerr effect, while the diffraction term, ψ_{xx}, is replaced, in the normalized form, by the groupvelocity dispersion (GVD), ±ψ_{ττ}, with + and − corresponding, respectively, to the anomalous and normal GVD, the former type being most relevant for optical fibers.
Seven years since the prediction by Hasegawa and Tappert (1973), the temporal solitons in fibers, both fundamental and higherorder ones (breathers), were created by Mollenauer et al. (1980). A copy of the record from that paper, which shows autocorrelation profiles of the temporal solitons, is displayed in Fig. 1.6. In this figure, a gradually decaying linear pulse is observed for a low input power, 0.3 W, which is followed by the formation of a fundamental soliton at power 5.0 W. At higher powers, viz., 11.4 and 22.5 W, the output splits, respectively, in two or three fundamental solitons. In the autocorrelation records, three or five peaks (only partly resolved, in the latter case) correspond, severally, to two or three distinct solitons.
The same NLS equation governs shaping of nonlinear beams of light in the spatial domain. In that case, the evolutional variable t in Eq. (1.9) is again replaced by the propagation distance, z, in a planar waveguide, while x is the transverse coordinate in the waveguide. The selffocusing cubic term accounts for the Kerr effect in the material of the waveguide, while ψ_{xx} represents paraxial diffraction of light in the guiding plane. Unlike the GVD, which may have either sign, the diffraction always has the sign that is tantamount to the anomalous GVD; hence, the spatialdomain NLS equation predicts the existence of spatial solitons in the form of selftrapped planar light beams. Experimentally, the spatial optical soliton was first created by Barthélémy et al. (1985) in a thin layer of a liquid nonlinear dielectric. Later on, the creation of spatial solitons was reported by Aitchison et al. (1990) in a planar waveguide made of glass and by Segev et al. (1992) in photorefractive (PhR) crystals, which exhibit saturable nonlinearity, instead of the cubic selffocusing in Eq. (1.9).
Other optical media may feature other nonlinearities, an important type being represented by the combination of selffocusing cubic and defocusing quintic terms. The corresponding version of the NLS equation is
where all coefficients may be set equal to 1 or −1 by means of scaling. While Eq. (1.31) with the cubic–quintic (CQ) nonlinearity is nonintegrable, its exact onesoliton solutions are available. They were first found by Pushkarov et al. (1979),
where ω takes values
The integral norm of the CQ soliton is
The local amplitude of these solutions cannot exceed $(\psi )max=3/2$. Accordingly, in the limit of ω → 3/16, the soliton takes a flattop (FT) shape, with constant amplitude $\psi \u22483/2$ and large width
All these solitons are stable.
At ω = −3/16, the bright soliton (1.32) goes over into a stationary front solution, which connects asymptotic flat states ψ = 0 and $\psi =(3/2)exp((3/16)it)$ (Birnbaum and Malomed, 2008),
The NLS equation with the CQ nonlinearity supports dark solitons as well, the difference from those in the cubic NLS equation [see Eq. (1.17)] being that they exist in the same finite interval of ω, given by Eq. (1.33), which is populated by the bright solitons and, in addition, in the adjacent semiinfinite interval, ω > 0,
Furthermore, in a narrow interval 3/16 < −ω < 1/4 adjacent to one (1.33) on its other side, Eq. (1.31) gives rise to the bubble solution (so called following Barashenkov and Makhankov, 1988), which features a local depression in a finite background, without crossing zero,
As shown in Chaps. 3 and 4, it has been demonstrated in many works, starting from one by QuirogaTeixeiro and Michinel (1997), that the CQ nonlinearity offers a very important setting for the creation of stable multidimensional solitons, including ones with embedded vorticity. There are diverse optical media that demonstrate the nonlinear response that may be approximated by the CQ combination, usually in a combination with conspicuous losses. An especially appropriate medium is provided by colloidal suspensions of metallic nanoparticles [see a review by Reyna and Araújo (2017)]. By adjusting the size and density of the nanoparticles, it is possible to produce a colloid whose nonlinear optical properties are very accurately fitted by the cubic and quintic susceptibilities while keeping losses at a relatively low level. Another appropriate medium is liquid carbon disulfide, in which stable (2+1)D solitons were created by Falcão Filho et al. (2013) [this notation implies that the spatial soliton is selftrapped in the transverse twodimensional plane, (x, y), while the third coordinate, z, plays the role of the evolutional variable].
1.1.2.3.2 Systems of coupled NLS equations
If the polarization of light in the optical fiber, represented by two complex amplitude fields ψ_{1,2}, is taken into regard, the NLS equation (1.9) is replaced by a system of two equations coupled by the crossphasemodulation (XPM), whose relative strength, g > 0, is defined with respect to the selfphase modulation (SPM), which accounts for the intrinsic nonlinearity of each polarization component,
[here, symbols t and x are kept for the evolutional variable and effective coordinate, and the anomalous sign of the fiber’s GVD is assumed, cf. Eq. (1.9)]. The most relevant values of the XPM/SPM coefficient in Eqs. (1.40) and (1.41) are g = 2/3 and g = 2, which correspond to the linear and circular polarizations, respectively (Agrawal, 2013). A similar system of XPMcoupled NLS equations with g = 2 models the copropagation of two waves carried by different wavelength in the fiber, or in a planar waveguide, in terms of the spatialdomain optics (Kivshar and Agrawal, 2003).
A wellknown fact is that the system of coupled NLS equations (1.40) and (1.41) is integrable, by means of the IST method, solely in the case of g = 1, as demonstrated by Manakov (1973). In that case, fundamentalsoliton solutions obviously reduce to ones produced by the single NLS equation, given by Eq. (1.10). In the nonintegrable system, with g ≠ 1, twocomponent fundamental solitons with unequal norms of the components, $\u222b\u2212\u221e+\u221e\psi 1,2(x)2dx$ (which are dynamical invariants of the system), can be found in a numerical form, or approximately by means of the VA (Kaup et al., 1993). It is relevant to stress that the existence of stationary singlesoliton solutions does not require integrability of the underlying equation(s). Further, the wellelaborated perturbation theory provides an efficient tool for the work with solitons governed by equations close to integrable ones (Kivshar and Malomed, 1989).
As shown by PérezGarcía and Beitia (2005) and Adhikari (2005), the XPMmediated attraction between the two components may support stable localized modes (called symbiotic solitons, as they cannot exist without the attraction between the components) in the case when the selfattraction in each component is repulsive. The modified form of Eqs. (1.40) and (1.41), respectively, is
with a real XPM coefficient g > 1. In fact, a similar system, with the intercomponent attraction slightly exceeding the intrinsic selfrepulsion, is necessary for the creation of quantum droplets, considered in detail below [see Eqs. (2.42) and (2.43)] (Petrov, 2015). A 2D version of symbiotic solitons was considered by Ma et al. (2016). Then, 1D and 2D symbiotic solitons whose components are additionally coupled by SOC were considered by Adhikari (2021). In the latter case, 2D solitons with the selfattractive cubic nonlinearity are stabilized against the critical collapse by the linear SOC terms, similar to how it was predicted by Sakaguchi et al. (2014).
Another system of NLS equations that plays an important role in optics is one including the linear coupling between the components, which model dualcore optical fibers or planar waveguides in the temporal or spatial domain, respectively (alias nonlinear couplers, Jensen, 1982),
with a real coupling coefficient K. Unlike the single NLS equation, this system is not integrable, while the Manakov system of Eqs. (1.40) and (1.41) with g = 1 remains integrable if the linear coupling is added to it, as shown by Tratnik and Sipe (1988). Unlike the system of Eqs. (1.40) and (1.41), the one based on Eqs. (1.42) and (1.43) conserves only the total norm,
but not the two norms separately.
Obvious symmetric soliton solutions of Eqs. (1.42) and (1.43), with ψ_{1} = ψ_{2}, become unstable against spontaneous symmetry breaking (SSB). Indeed, the consideration of infinitely small antisymmetric perturbations added to the symmetric soliton demonstrates the onset of the SSB instability at
(Wright et al., 1989). Past the SSB point, the unstable symmetric solitons are replaced by stable asymmetric ones.
The SSB scenario is accurately outlined by the VA based on the straightforward Ansatz,
cf. Eq. (1.23). In this Ansatz, parameter θ determines the asymmetry of the soliton,
The transition from the symmetric solitons to asymmetric ones is summarized by the bifurcation diagram shown in Fig. 1.5(b), which shows the asymmetry measure (1.47) as a function of the total norm (1.44). The diagram is produced by the VA, with its counterpart generated by the numerical solution of Eqs. (1.42) and (1.43) being very close to the picture displayed in the figure. In particular, the VA predicts the bifurcation point,
the comparison of which with the exact value (1.45) shows that the relative error of the VA is ≈6%. The diagram shown in Fig. 1.5(b) includes two mutually symmetric branches corresponding to the obvious swap of the components, ${1\u21c42}$. Note that the bifurcation is of the subcritical type, with the asymmetric states appearing at values of the total norm slightly smaller than the bifurcation point,
cf. the value of the total norm at the bifurcation point, which is given by Eq. (1.48).
Many aspects the topic of SSB in twocomponent nonlinear systems are comprised by the volume edited by Malomed (2013).
1.1.2.3.3 The Bragggrating (BG) model
In fiber optics, the GVD, which is necessary for the creation of bright solitons, is provided by material properties of silica glass. Alternatively, it may be created artificially by means of a Bragg grating (BG) written in the cladding of the fiber. The BG is an array of local defects, with the period equal to half the wavelength of the carrier electromagnetic wave. Under this condition, the resonant Bragg scattering transforms right and lefttraveling waves, ψ(x, t) and ϕ(x, t), into each other. In the normalized form, the corresponding system of propagation equations, which takes the SPM and XPM nonlinearity into regard, is written as [see details in the review by de Sterke and Sipe (1994)]
This system is not integrable, but a full family of its onesoliton solutions was found by Christodoulides and Joseph (1989) and Aceves and Wabnitz (1989), in a rather cumbersome form,
where parameters δ (0 < δ < π) and c (−1 < c < +1) determine the soliton’s amplitude and velocity. Note that, although the velocity is limited to interval c < 1, similar to the speed of relativistic particles, Eqs. (1.50) and (1.51) are not invariant with respect to the Lorentz transform, therefore obtaining moving solitons from quiescent ones is a nontrivial result, unlike the application of the Galilean boost (1.13) to the NLS soliton (1.10). These solutions are called gap solitons (GSs) because the dispersion relation for the planewave solutions, ${\psi ,\varphi}\u223cexp(ikx\u2212i\omega t)$, of the linearized version of Eqs. (1.50) and (1.51),
includes a gap, ω < 1, in which the solitons exist. The concept of GSs was introduced by Chen and Mills (1987). Mathematically, rigorous treatment of this topic is provided in the book by Pelinovsky (2011).
Analysis of stability of the GS family given by Eqs. (1.52)–(1.55) produces nontrivial results. For the first time, a possibility of the existence of a stability boundary inside the family was reported by Malomed and Tasgal (1994) by means of the VA. Accurate results were then produced by Barashenkov et al. (1998) and De Rossi et al. (1998). The conclusion is that at c = 0, the quiescent GSs are stable in interval 0 < δ < δ_{cr} ≈ 0.505π, i.e., almost exactly in the half of their existence domain (0 < δ < π). At c ≠ 0, the dependence of δ_{cr} on c is very weak.
Integral characteristics of the BG modes are the total norm and momentum, which are dynamical invariants of the BG model based on Eqs. (1.40) and (1.41) [cf. Eq. (1.17) for the NLS equation]. Their values for the GSs are
Simulations of Eqs. (1.50) and (1.51) demonstrate that collisions between slow solitons, with velocities $c\u22720.2$, give rise to their merger into a single bound state (Mak et al., 2003), while faster ones pass through each other.
In the experiment, moving GSs in the fiber BG were first created by Eggleton et al. (1996). A challenging possibility is to create quiescent or very slow BGs with c = 0 or small c, as it would be a fascinating phenomenon, exhibiting pulses of “standing light.” The slowest GS that was reported in the experiment of Mok et al. (2006) had the velocity ≈1/7 of the light speed in vacuum.
Also relevant to optics are models of BGs embedded in a dissipative nonlinear medium. In particular, such models predict specific soliton states (Tran and Rosanov, 2008).
1.1.2.4 Physical realizations of the NLS equation: Bose–Einstein condensates (BECs)
1.1.2.4.1 Basic models for the dynamics of matter waves in free space
The NLS equation (1.9) is tantamount to the Gross–Pitaevskii (GP) equation for the wave function of the BEC, derived in the meanfield (MF) approximation (Pitaevskii and Stringari, 2003; and Pethick and Smith, 2008). The original GP equation with the cubic nonlinearity is derived in 3D, and the natural sign of the cubic term corresponds to the selfrepulsion of the wave function, as this term represents effects of interatomic collisions, with atoms bouncing back from each other. Nevertheless, the sign of the interaction, as well as its strength, can be readily changed by means of uniform dc magnetic field applied to the condensate. The magnetic field gives rise to the Feshbach resonance (FR), i.e., the formation of a quasibound state by the colliding atoms, which strongly affects the resulting scattering length (Bradley et al., 1995; and Chin et al., 2010). The interaction becomes attractive if the FR makes the effective scattering length negative, which was realized, in particular, in the gases of ^{85}Rb (Cornish et al., 2006), ^{39}K (Roati et al., 2007), and ^{7}Li (Pollack et al., 2009) atoms.
It is necessary to stress that the GP equation in the form of Eq. (1.9) pertains, strictly speaking, to BEC at zero temperature. Effects of finite temperatures are taken into regard by adding stochastic and dissipative terms to the GP equation (Proukakis et al., 2013). These aspects of the BEC theory are not addressed in the present book.
The dimensional reduction, 3D → 1D, with the remaining free coordinate x, is performed by assuming the action of a tightly confining potential in the transverse plane, (x, y). The resulting GP/NLS equation for lowdensity condensates keeps the form of Eq. (1.9). For denser ones with the attractive selfinteraction, the dimensional reduction leads to the onedimensional NLS equation with nonpolynomial nonlinearity, unlike the CQ nonlinearity in Eq. (1.31). It was derived by Salasnich et al. (2002a), taking into regard the variation of the transverse confinement radius of the condensate. The normalized form of the equation is
with a selfinteraction coefficient γ > 0. In this notation, the 1D equation with γ > 1 develops a singularity at a critical value of the 1D density, ψ^{2} = 1. In fact, the supercritical collapse is a fundamental property of the threedimensional GP equation with the cubic selfattraction (Bergé, 1998; Sulem and Sulem, 1999; Zakharov and Kuznetsov, 2012; and Fibich, 2015), as discussed in detail below. Accordingly, the presence of the singularity in the 1D equation (1.59) is an effective 1D reduction of the 3D collapse (Salasnich et al., 2002b). A detailed analysis of the connection between the full 3D GP equation and the 1D reduction close to the collapse threshold was performed by Cuevas et al. (2013).
In the case of the selfrepulsive cubic nonlinearity in the threedimensional GP equation, the reduction to 1D leads to another type of the nonpolynomial nonlinearity, as demonstrated by Muñoz Mateo and Delgado (2008). In a normalized form, this equation is
This equation provides an accurate approximation in comparison with the full 3D solution—in particular, for gap solitons sustained in the selfrepulsive BEC under the action of a tight cigarshaped trapping potential, combined with a spatially periodic opticallattice potential applied in the axial direction (Muñoz Mateo et al., 2011).
1.1.2.4.2 The Gross–Pitaevskii equation for matter waves and gap solitons under the action of opticallattice (OL) potentials
Getting back to the basic GP equation in 1D, it is necessary to note, once again, that experimental setups include potentials acting in the longitudinal direction. Accordingly, Eq. (1.9) is replaced by the equation including a potential, U(x),
An important type of the potential is a spatially periodic one, with depth 2V_{0} and period L (by means of rescaling, one can fix the value of L, which is often scaled to be L = π),
This potential represents the optical lattice (OL), which may be induced, as an interference pattern, by laser beams driving the nearresonant dipole interaction with atoms. The physics of BEC under the action of OL potentials was reviewed by Morsch and Oberthaler (2006).
Similar to the GP equation for BEC including the OL potential is the NLS equation for optical beams propagating in photonic crystals (PhCs), i.e., waveguiding media with periodic modulation of the refractive index (and possibly, of the Kerr coefficient) along the transverse coordinates. The concept of PhCs was introduced by Yablonovitch (1993) [see also books by Joannopoulos et al. (2008), Skorobogatiy and Yang (2009), and Yang (2010)]. A difference from potential (1.62) is that a realistic form of an effective periodic potential in PhC, induced by the transverse modulation of the local refractive index, is of the Kronig–Penney type, i.e., a piecewiseconstant one [Kronig and Penney (1931); see also a review by Hennig and Tsironis (1999)].
An important property of Eq. (1.61) with potential (1.62) is the bandgap spectrum of Eq. (1.61). It is produced by the substitution of an obvious expression for stationary eigenmodes,
with real chemical potential μ and real function ϕ(x), in Eq. (1.61),
The spectrum is calculated numerically from the solution of the linearized version of Eq. (1.64) [see the book by Yang (2010)].
It addition to the usual semiinfinite gap, which extends to μ → −∞, the bandgap spectrum contains an infinite set of finite gaps, in which linear states (Bloch modes) do not exist, which makes it possible to look for GSs (gap solitons) populating finite gaps or parts thereof. An example of the spectrum is displayed in the left panel of Fig. 1.7. A remarkable peculiarity of this system is that GSs are sustained by the repulsive sign of the nonlinearity. This counterintuitive finding is explained by the fact that the bandgap spectrum can make the effective mass of excitation modes in the periodic potential negative; hence, its interplay with the repulsive nonlinearity produces, roughly speaking, the same results (selftrapping of bright solitons) as the combination of the normal positive effective mass and selfattraction [Abdullaev et al., 2001; and Carusotto et al., 2002; see also a review by Brazhnyi and Konotop (2004)].
The presence of the periodic potential makes it possible to combine GSs of the fundamental type into higherorder bound states. Examples of the GSs obtained in the first and second bandgaps of the spectrum of the OL potential (1.62) are displayed on Figs. 1.8 and 1.9, respectively. Families of the GS solutions are naturally presented, in the right panel of Fig. 1.7, by dependences of their norm versus the chemical potential,
In addition to the fundamental GSs and their bound states, Eq. (1.64) gives rise to subfundamental GSs (branch F in the right panel of Fig. 1.7 and in Fig. 1.9), which are, essentially, odd (spatially antisymmetric) modes squeezed into a single cell of the OL potential (Mayteevarunyoo and Malomed, 2006). The name indicates that, for given μ, the norm of these states is lower than that of the fundamental GSs. Indeed, in the right panel of Fig. 1.7, the subfundamental branch F runs below the fundamental one, E.
Most GS families are stable, except for the subfundamental branch, which is completely unstable (as shown in the right panel of Fig. 1.7), as well as all bound states including subfundamental GSs (Mayteevarunyoo and Malomed, 2006). A detailed analysis of the stability of GS families was given by Kizin et al. (2016).
Although Figs. 1.8 and 1.9 suggest that GSs are strongly pinned to the underlying lattice, direct numerical simulations of Eq. (1.61) with OL potential (1.62), initiated by the application of kick k to a quiescent soliton, i.e., with initial conditions,
demonstrate mobility of the GSs, with a negative effective dynamical mass. As a result, if the additional harmonicoscillator (HO) potential is added to Eq. (1.61),
simulations demonstrate that such a trapping potential, with Ω^{2} > 0, expels the GS, while an expulsive potential, with Ω^{2} < 0, plays the role of a trap for the negativemass GS—in particular, it performs stable oscillatory motion in the latter case (Sakaguchi and Malomed, 2004).
1.1.2.4.3 Experimental demonstration of fundamental solitons, breathers, and gap solitons in BEC
In BEC experiments, quasi1D dark matterwave solitons were made first in the condensate of ^{87}Rb atoms, by means of the phaseimprinting technique (Burger et al., 1999; and Denschlag et al., 2000). Next, the creation of fundamental bright solitons in the ultracold gas of ^{7}Li loaded in a quasi1D cigarshaped trapping potential, with the interaction switched to weak selfattraction by means of the FR, was reported independently by Strecker et al. (2002) and Khaykovich et al. (2002) [see also a review by Strecker et al. (2003)]. The former work produced, under controllable conditions, not only isolated solitons but also soliton clusters [see an example in Fig. 1.10(a)]. Collisions between the matterwave solitons were also studied experimentally and theoretically (Nguyen et al., 2014). As a result, various outcomes of the collisions were identified, such as rebound or passage of the solitons, as well as their merger and collapse (blowup). Using a different experimental technique, similar solitons were later created by Medley et al. (2014).
Bright solitons were also made in the condensate of ^{85}Rb atoms by Cornish et al. (2006). In this case, the shape of the solitons was nearly spherical, but, nevertheless, they were quasi1D dynamical states, in the sense that the nonlinear selfattraction accounted for the selftrapping in the longitudinal direction, while the transverse confinement was imposed by the trapping potential. Later, solitons in the same atomic species were used by Marchant et al. (2013) for the experimental study of interactions with potential barriers [see Fig. 1.10(b)] and by McDonald et al. (2014) for the operation of a soliton interferometer. Further, the creation of quasi1D bright solitons was reported by Lepoutre et al. (2016) in the condensate of ^{39}K atoms, and by Mežnaršič et al. (2019) in ^{133}Cs.
Higherorder solitons (matterwave breathers) in BEC were created more recently by means of the quench of the strength of the attractive nonlinearity. In particular, starting from an established fundamental soliton, the FR was used to suddenly multiply the nonlinearity strength by a factor of 4. In terms of the GP equation (1.9) with the top sign, this is tantamount to suddenly multiplying ψ by $N=2$, which, in turn, implies the conversion of the fundamental soliton into the the secondorder breather, as per Eq. (1.15). Similarly, the quench amounting to the multiplication of the nonlinearity strength by 9 is tantamount to the sudden conversion of the fundamental soliton into the thirdorder breather. Using this method, breathers with $N=2$ were experimentally created in the ultracold gas of ^{133}Cs by Di Carli et al. (2019) and in the gas of ^{7}Li by Luo et al. (2020). In the latter work, breathers with $N=3$ were created too. The second and thirdorder breathers produced in that work are displayed in Fig. 1.11. It is seen in panel (a) that the breather with $N=2$ performs periodic oscillations of its width, keeping the peak density at the center, cf. the exact shape of the breather solution in Fig. 1.3(b). Further, Fig. 1.11(b) clearly shows that the thirdorder breather periodically splits into two separated density peaks and recombines back into a single one, cf. the exact solution for $N=3$ presented in Fig. 1.8(b).
As concerns GSs (gap solitons), direct observation of such modes in the selfrepulsive condensate (^{87}Rb) loaded into a trap with an OL potential was reported by Eiermann et al. (2004). In that experiment, the GS was built of ca. 250 atoms. This number was sufficient to take well resolved images of the GSs. In PhCs (photonic crystals), the creation of optical GSs is a less essential issue, because, in optics, the cubic Kerr nonlinearity is normally selffocusing; hence, it can create spatial quasi1D solitons without the help of a periodic potential. However, a possibility to make GSs is quite essential in another area of photonics, which deals with coherent exciton–polariton fields in semiconductor microcavities [a review of the topic was provided by Byrnes et al. (2016)]. In this system, the nonlinearity is defocusing (due to repulsive interactions between parallel electric dipole moments of excitons). The creation of GSs in the exciton–polariton condensate, under the action of a spatially periodic structure built into the embedding microcavity (which plays the role of the PhC in this setup), was reported by CerdaMéndez et al. (2013).
1.1.3 The sineGordon equation
In addition to the KdV and NLS equations, the third commonly known integrable 1D equation is the sineGordon (SG) equation for real function ϕ(x, t),
Another frequently employed form of the SG equation is provided by the use of the lightcone coordinates, ξ = (x + t)/2, τ = (x − t)/2,
In the form of Eq. (1.69), the integrability of the SG equation was discovered in the nineteenth century by Swedish mathematician Bäcklund (1876), which definitely makes it the oldest member of the family of integrable nonlinear partial differential equations. The integrability was originally found in the form of what is commonly called the Bäcklund transform. Being applied to a known nsoliton solution, it generates a solution composed of (n + 1) solitons. Thus, starting from the zero solution, ϕ = 0, which corresponds to n = 0, one can construct an infinite set of exact solutions. The integrability of the SG equation in the framework of the IST method was demonstrated later (Ablowitz et al., 1974; and Zakharov et al., 1974). A link between the Bäcklund transform and IST is a wellknown ingredient of the integrability [see the book by Rogers and Schief (2002)].
In physics, the SG equation had appeared as the continuum limit of the celebrated dynamical model for dislocations in crystals, which was introduced by Frenkel and Kontorova (1939). The FK model amounts to a set of equations of motion for coordinates u_{n}(t) of a string of atoms in a crystal lattice [see a detailed presentation in the book by Braun and Kivshar (2004)],
with a coupling constant C > 0. A similar model for dislocations was introduced by Frank and der Merwe (1949). Later, the SG equation in the form of Eq. (1.68) had found a very important realization as the model of long Josephson junctions (JJs), i.e., narrow dielectric layers separating bulk superconductors (Josephson, 1962). In this case, ϕ(x, t) in Eq. (1.68) is the local phase jump between macroscopic wave functions of superconducting electrons across the junction. Long JJs admit the existence of topological solitons in the form of fluxons [quanta of the magnetic flux trapped in the junction, represented below by Eq. (1.72)], which were created in the experiment by Fulton and Dynes (1973). Experimental and theoretical studies of JJs, and, especially, of long ones have grown into a vast research area [see books by Barone and Paternó (1982) and Ustinov (2015)]. A review of applications of the SG equation can be found in the article by Malomed (2014).
The SG and NLS equations are related by a possibility to look for approximate smallamplitude broad solutions to Eq. (1.68) in the asymptotic form of
where ψ is a slowly varying complex function subject to constraint ψ^{2} ≪ 1 (recall $*$ stands for the complex conjugate). In the lowest asymptotic approximation, the substitution of Ansatz (1.71) in Eq. (1.68) and appropriate expansion leads to the NLS equation (1.9) with the top sign.
Elementary exact solutions of Eq. (1.68) are kinks (+) and antikinks (−), alias fluxons, in terms of the JJs, traveling with velocity c, which may take values in a “relativistic” interval, −1 < c < +1,
Obviously, moving kinks can be generated by the Lorentz transform from the quiescent one (with c = 0), although, in physical units, the largest velocity admitted by the SG equation modeling the long JJ (the Swihart velocity) is usually ∼0.01 of the light speed in vacuum (Barone and Paternó, 1982). The kinks and antikinks are usually considered as the simplest example of topological solitons, with the corresponding topological charge (alias polarity) defined as [ϕ_{kink}(x = +∞) − ϕ_{kink}(x = −∞)]/(2π) = ±1. Note that, although the phase field represented by kink (1.72) is not localized (taking value 2π at x = ±∞), the respective energy (Hamiltonian) and momentum densities corresponding to the SG equation,
are well localized, making the total energy and momentum of the kink convergent,
A nontrivial exact solution of Eq. (1.68), provided by the exact integrability, is a breather, i.e., an oscillatory kink–antikink bound state, with zero topological charge,
which is plotted in Fig. 1.12. The family of solutions (1.76) is parameterized by constant θ taking values 0 < θ < π/2. The energy of the breather is
In the limit of θ ≪ π/2, substitution (1.71) makes the breather asymptotically equivalent to the NLS soliton (1.10) with c = 0. Breathers moving at velocity c can be produced by the application of the Lorentz boost (transform) to solution (1.76), which has c = 0.
Another nontrivial solution of the SG equation describes kink–kink and kink–antikink collisions, which are displayed in Fig. 1.13. The plots clearly demonstrate that kinks with identical polarities repel each other; hence, they bounce back as a result of the collision, while kinks with opposite polarities interact attractively, passing through each other in the course of the collision. In particular, the kink–antikink collision with zero velocities at infinitely large separation between them is given by the limit form of the breather solution (1.76) corresponding to θ → π/2,
Irrespective of the relative polarity of the kinks colliding with velocities ±c, the effect of the interaction is a shift of each kink in the direction of its motion by
cf. the abovementioned similar exact results for the shift of colliding KdV and NLS solitons.
1.1.4 Discrete solitons in lattice media
An important ramification of the experimental and theoretical work on spatial optical solitons is the creation of such modes in arrays of waveguides, which is a subject of discrete nonlinear optics—a vast research area dealing with many species of discrete solitons [see a review by Lederer et al. (2008)]. The fundamental model for discrete solitons is provided by the discrete NLS equation,
where the discrete coordinate n replaces the continuous one x in Eq. (1.9), and the finitedifference combination replaces the second derivative ψ_{xx}. The realization of Eq. (1.80) in an array of linearly coupled waveguides was proposed by Christodoulides and Joseph (1988) and Aceves et al. (1994).
Unlike its continuum counterpart, where signs ± correspond to two different equations, Eq. (1.80) with the bottom sign in front of the nonlinear term can be transformed into its counterpart with the top sign, for discrete wave field $\psi ~n$, by means of the staggering transformation (Cai et al., 1994),
Studies of discrete solitons in this model have produced a great body of results, which were summarized, in particular, in the book of Kevrekidis (2009) and more recently by Malomed (2020). Discrete optical solitons, which represent fundamental solutions of Eq. (1.80), were created experimentally by Eisenberg et al. (1998) and Morandotti et al. (1999).
It is necessary to stress that the discrete NLS equation (1.80), although originating from the integrable NLS equation (1.9), is nonintegrable [similarly, the discretization of the integrable SG equation leads to the nonintegrable FKlattice equation (1.70)]. In particular, instead of the abovementioned infinite set of dynamical invariants sustained by Eq. (1.9), Eq. (1.80) conserves only two invariants, viz., the norm and Hamiltonian,
Discretesoliton solutions of Eq. (1.80) are looked for in a straightforward form,
with real μ < 0 (the chemical potential, in terms of the GP equation) and real discrete field u_{n} satisfying equation
These solutions are not available in an exact analytical form, but Eq. (1.86) can be easily solved numerically, and the solutions can be approximated analytically by means of the variational method (VA), noticing that the equation can be derived by the variation of the corresponding Lagrangian,
One can define two types of discrete solitons, viz., onsite (OS)centered and intersite (IS)centered ones, which are distinguished by their symmetry,
Thus, the OScentered solitons have a single peak at n = 0, while their IScentered counterparts have equal maxima at two points, n = 0 and n = 1 (see Fig. 1.14).
The VA for these two species of the discrete solitons was elaborated by Malomed and Weinstein (1996) and Kaup (2005), based on Ansätze,
for the modes of the OS and IScentered types, respectively. In particular, the substitution of Ansatz (1.90) in Lagrangian (1.87) yields
which leads to the Euler–Lagrange equations, ∂L^{(VA)}/∂(A^{2}, a) = 0. As a result, the VA predicts discrete solitons of both types, which, for moderately strong discreteness [i.e., a not too small in Ansätze (1.78) and (1.79)], are very close to their numerically found counterparts, as shown in Fig. 1.14.
While the discrete NLS equation (1.68) is not integrable, an integrable discretization of the NLS equation was discovered by Ablowitz and Ladik (1976),
where positive and negative values of the real nonlinearity coefficient, λ, correspond to the selffocusing and defocusing, respectively [for a detail consideration of the compatibility of discretization and integrability, see the book by Suris (2003)]. Considerable interest was also drawn to a nonintegrable combination of the Ablowitz–Ladik (AL) and discrete NLS equations, which is known as the Salerno model (Salerno, 1992),
The AL and Salerno models do not have straightforward physical realizations. Nevertheless, the latter one appears as a semiclassical limit of the Bose–Hubbard model, which represents quantum BEC loaded in a deep OL potential, in the case when dependence of the hopping rate between adjacent cells of the OL on populations of the cells is taken into regard (Dutta et al., 2015).
These models conserve the total norm, but its definition is different from the straightforward one, given by Eq. (1.70) for the discrete NLS equation:
The Hamiltonian of these models is also essentially different from the discreteNLS Hamiltonian given by Eq. (1.83): for the AL model,
and for the Salerno model, it is (Cai et al., 1996)
with the last term given by Eq. (1.95). The price paid for the “simplicity” of Hamiltonians (1.96) and (1.97) is the complex form of the respective Poisson brackets, which determine the equations of motion in terms of the Hamiltonian as dψ_{n}/dt = {H, ψ_{n}}. For both the AL and Salerno models, the brackets, written for a pair of functions of the discrete dynamical variables, B(ψ_{n}, ψ_{n}*) and C(ψ_{n}, ψ_{n}*), are
The AL equation (1.93) gives rise to an exact solution for solitons in the case of the selffocusing nonlinearity, λ > 0. Setting, in this case, λ = 1 by means of rescaling, the solution is
where β and α are arbitrary real parameters that determine the soliton’s amplitude, sinh β, its velocity, V = 2β^{−1}(sinh β) sin α, and frequency Ω = −2[(cosh β) cos α + (α/β)(sinh β) sin α]. The existence of exact solutions for traveling solitons in the discrete system is a highly nontrivial property, which is a manifestation of the integrability of the AL equation. If the system is not integrable, the motion of a discrete soliton through the lattice is impeded by emission of radiation (lattice phonons), even if, in many cases, this effect may seem very weak in direct simulations (Duncan et al., 1993). The Salerno model gives rise to species of (numerically found) stationary discrete solitons, which are different for λ > 0 and λ < 0 (GómezGardeñes et al., 2006).
It is relevant to mention a class of semidiscrete optical systems, built of two coupled waveguiding layers, one continuous and one effectively discrete. An example is a discrete array mounted on top of a slab. As shown by Panoiu et al. (2008), the setup is modeled by the system of discrete and continuous NLS equations for the discrete and continuous optical amplitudes, ϕ_{n}(z) and ψ(x, z), which are nonlinearly coupled by the XPM interaction with the respective real coefficient κ [cf. the system of Eqs. (1.40), (1.41)],
Here, z is the propagation distance, n represents the discrete coordinate in the array, x is the transverse coordinate in the slab, and β is a propagationconstant mismatch between the slab and the array. This system gives rise to stable semidiscrete solitons.
Semidiscrete systems with the quadratic (χ^{(2)}) nonlinearity, composed of a discrete equation for the fundamentalfrequency (FF) wave and a continuous equation for the second harmonic (SH), or vice versa, were introduced by Panoiu et al. (2006). These systems also produce stable semidiscrete solitons.
1.2 Exit to the Multidimensional World
1.2.1 The first step: Twodimensional Townes solitons
While there are important setups that make it possible to introduce physically relevant 1D systems, as discussed above, the real world is threedimensional or, in some cases, quasitwodimensional. This obvious fact strongly suggests to consider 3D and 2D solitons in nonlinear optics, BEC, plasmas, and other nonlinear physical media.
As the simplest relevant model, which admits direct extension from 1D to 3D and 2D, is the cubic NLS/GP equation [in fact, as mentioned above, its 1D version (1.9) was derived by the inverse reduction, 3D → 1D],
where $\u22072$ is the 3D or 2D Laplacian, and the sign in front of the cubic term corresponds to the attractive nonlinearity. The Hamiltonian corresponding to Eq. (1.100) is
where $\u222bdDx$ stands for the integration in the 3D or 2D space.
In the realization in optics, the evolutional variable t in Eq. (1.100) is replaced by the propagation distance, z, and one of the transverse coordinates is replaced by the local time defined in Eq. (1.30′) (like in the onedimensional NLS equation for optical fibers), while two other spatial variables, (x, y), keep the meaning of the transverse coordinates in the bulk optical waveguide.
Fundamental (isotropic) localized solutions of Eq. (1.100) are looked for in the usual form [cf. Eqs. (1.63) and (1.85)],
where μ < 0 is a real chemical potential, r is the radial coordinate, and real function ϕ(r) obeys the equation
where D = 2 or 3 is the spatial dimension. Obviously, localized solutions of Eq. (1.103) have the asymptotic form at large r
The general Ansatz for vortex modes in the 2D geometry is
where θ is the azimuthal coordinate and integer S is the winding number (alias vorticity or topological charge) (Pismen, 1999). The substitution of Ansatz (1.105) in Eq. (1.100) gives rise to 2D solitons with winding numbers S = ±1, ± 2,…(Kruglov and Vlasov, 1985; Kruglov et al., 1988; and Kruglov et al., 1992). In this case, Eq. (1.103) for real function ϕ_{S}(r) is replaced by