Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients

Shortcuts to adiabatic soliton compression in active nonlinear Kerr media

We implement variational shortcuts to adiabaticity for optical pulse compression in an active nonlinear Kerr medium with distributed amplification and spatially varying dispersion and nonlinearity. Starting with the hyperbolic secant ansatz, we employ a variational approximation to systematically derive dynamical equations, establishing analytical relationships linking the amplitude, width, and chirp of the pulse. Through the inverse engineering approach, we manipulate the distributed gain/loss, nonlinearity and dispersion profiles to efficiently compress the optical pulse over a reduced distance with high fidelity. In addition, we explore the dynamical stability of the system to illustrate the advantage of our protocol over conventional adiabatic approaches. Finally, we analyze the impact of tailored higher-order dispersion on soliton self-compression and derive physical constraints on the final soliton width for the complementary case of soliton expansion. The broader implications of our findings extend beyond optical systems, encompassing areas such as cold-atom and magnetic systems highlighting the versatility and relevance of our approach in various physical contexts.

Self-similar pulse compression in a tapered Pb-silicate photonic crystal fiber at 2 µm

We report a 2-µm all-fiber nonlinear pulse compressor based on a tapered Pb-silicate photonic crystal fiber (PCF), which is capable of achieving large compression with low pedestal energy. A tapered Pb-silicate photonic crystal fiber with increased nonlinear coefficients is proposed for achieving self-similar pulse compression (SSPC) at 2 µm. The dynamic evolution of the fundamental order soliton is numerically analyzed based on the designed tapered fiber. After pulse compression in a tapered fiber with a length of 2.2 m, an initial 1.76 ps pulse can be compressed to 88 fs, increasing the peak power from 4.4 to 86 W with a compression factor of 20 and a quality factor of 98%. The results reveal that exponential variation yields superior compression performance and provides a promising solution for generating high-power femtosecond pulses at 2 µm.

Supercontinuum in integrated photonics: generation, applications, challenges, and perspectives

Frequency conversion in nonlinear materials is an extremely useful solution to the generation of new optical frequencies. Often, it is the only viable solution to realize light sources highly relevant for applications in science and industry. In particular, supercontinuum generation in waveguides, defined as the extreme spectral broadening of an input pulsed laser light, is a powerful technique to bridge distant spectral regions based on single-pass geometry, without requiring additional seed lasers or temporal synchronization. Owing to the influence of dispersion on the nonlinear broadening physics, supercontinuum generation had its breakthrough with the advent of photonic crystal fibers, which permitted an advanced control of light confinement, thereby greatly improving our understanding of the underlying phenomena responsible for supercontinuum generation. More recently, maturing in fabrication of photonic integrated waveguides has resulted in access to supercontinuum generation platforms benefiting from precise lithographic control of dispersion, high yield, compact footprint, and improved power consumption. This Review aims to present a comprehensive overview of supercontinuum generation in chip-based platforms, from underlying physics mechanisms up to the most recent and significant demonstrations. The diversity of integrated material platforms, as well as specific features of waveguides, is opening new opportunities, as will be discussed here.

Analytical Method for Generalized Nonlinear Schrödinger Equation with Time-Varying Coefficients: Lax Representation, Riemann-Hilbert Problem Solutions

In this paper, a generalized nonlinear Schrödinger (gNLS) equation with time-varying coefficients is analytically studied using its Lax representation and the associated Riemann-Hilbert (RH) problem equipped with a symmetric scattering matrix in the Hermitian sense. First, Lax representation and the associated RH problem of the considered gNLS equation are established so that solution of the gNLS equation can be transformed into the associated RH problem. Secondly, using the solvability of unique solution of the established RH problem, time evolution laws of the scattering data reconstructing potential of the gNLS equation are determined. Finally, based on the determined time evolution laws of scattering data, the long-time asymptotic solution and N-soliton solution of the gNLS equation are obtained. In addition, some local spatial structures of the obtained one-soliton solution and two-soliton solution are shown in the figures. This paper shows that the RH method can be extended to nonlinear evolution models with variable coefficients, and the curve propagation of the obtained N-soliton solution in inhomogeneous media is controlled by the selection of variable–coefficient functions contained in the models.

Chirped Periodic and Solitary Waves for Improved Perturbed Nonlinear Schrödinger Equation with Cubic Quadratic Nonlinearity

In this paper, we study the improved perturbed nonlinear Schrödinger equation with cubic quadratic nonlinearity (IPNLSE-CQN) to describe the propagation properties of nonlinear periodic waves (PW) in fiber optics. We obtain the chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF) and also obtain some solitary waves (SW) such as dark, bright, hyperbolic, singular and periodic solitons. The nonlinear chirp associated with each of these optical solitons was observed to be dependent on the pulse intensity. The graphical behavior of these waves will also be displayed.

Coupled Helmholtz equations: Chirped solitary waves

We investigate the existence and stability properties of chirped gray and anti-dark solitary waves within the framework of a coupled cubic nonlinear Helmholtz equation in the presence of self-steepening and a self-frequency shift. We show that for a particular combination of self-steepening and a self-frequency shift, there is not only chirping but also chirp reversal. Specifically, the associated nontrivial phase has two intensity dependent terms: one varies as the reciprocal of the intensity, while the other, which depends on non-Kerr nonlinearities, is directly proportional to the intensity. This causes chirp reversal across the solitary wave profile. A different combination of non-Kerr terms leads to chirping but no chirp reversal. The influence of a nonparaxial parameter on physical quantities, such as intensity, speed, and pulse width of the solitary waves, is studied as well. It is found that the speed of the solitary waves can be tuned by altering the nonparaxial parameter. Stable propagation of these nonparaxial solitary waves is achieved by an appropriate choice of parameters.

Dependence of the nonlinear behaviour stability of a ZnO nanoparticle-Bi2 O3 -Mn2 O3 -based varistor system on cooling rates during ceramic processing

A conventional ceramic processing method was applied to manufacture high-density 20-nm ZnO-Bi₂ O₃ -Mn₂ O₃ varistor ceramics. Different cooling rates in the range of 135-540°C/h led to a relatively slight influence on the microstructure, varistor voltage, and leakage current. In contrast, these rates strongly affected the nonlinear exponent. The specific surface area of the 20-nm ZnO nanoparticle may have led to an intense solid-state reaction even at a low cooling rate. Superior nonlinearity, with 59.7 μA nonlinear current leakage and 273.5 μA leakage current, was achieved at the 135°C/h cooling rate. The differences in the cooling rates led to a remarkable change in the material's stability under direct current (DC)-accelerated aging stress in the following order: 135°C/hr ˃ 270°C/hr˃405°C/hr ˃ 540°C/hr.

Shortcuts to Adiabaticity for Optical Beam Propagation in Nonlinear Gradient Refractive-Index Media

In recent years, the concept of “shortcuts to adiabaticity" has been originally proposed to speed up sufficiently slow adiabatic process in various quantum systems without final excitation. Based on the analogy between classical optics and quantum mechanics, we present a study on fast non-adiabatic compression of optical beam propagation in nonlinear gradient refractive-index media by using shortcuts to adiabaticity. We first apply the variational approximation method in nonlinear optics to derive the auxiliary equation for connecting the beam width with the refractive index of the medium. Then, the gradient refractive index is inversely designed through the perfect compression of beam width with the appropriate boundary conditions. Finally, the comparison with conventional adiabatic compression is made, showing the advantage of our shortcuts.

Propagation of dipole solitons in inhomogeneous highly dispersive optical-fiber media

We consider ultrashort light pulse propagation through an inhomogeneous monomodal optical fiber exhibiting higher-order dispersive effects. Wave propagation is governed by a generalized nonlinear Schrödinger equation with varying second-, third-, and fourth-order dispersions, cubic nonlinearity, and linear gain or loss. We construct a type of exact self-similar soliton solution that takes the structure of a dipole via a similarity transformation connected to the related constant-coefficients one. The conditions on the optical-fiber parameters for the existence of these self-similar structures are also given. The results show that the contribution of all orders of dispersion is an important feature to form this kind of self-similar dipole pulse shape. The dynamic behaviors of the self-similar dipole solitons in a periodic distributed amplification system are analyzed. The significance of the obtained self-similar pulses is also discussed. By performing numerical simulations, the self-similar soliton solutions are found to be stable under slight disturbance of the constraint conditions and the initial perturbation of white noise.

Chirped self-similar solitary waves for the generalized nonlinear Schrödinger equation with distributed two-power-law nonlinearities

We investigate the propagation characteristics of the chirped self-similar solitary waves in non-Kerr nonlinear media within the framework of the generalized nonlinear Schrödinger equation with distributed dispersion, two-power-law nonlinearities, and gain or loss. This model contains many special types of nonlinear equations that appear in various branches of contemporary physics. We extend the self-similar analysis presented for searching chirped self-similar structures of the cubic model to a more general problem involving two nonlinear terms of arbitrary power. A variety of exact linearly chirped localized solutions with interesting properties are derived in the presence of all physical effects. The solutions comprise bright, kink and antikink, and algebraic solitary wave solutions, illustrating the potentially rich set of self-similar pulses of the model. It is shown that these optical pulses possess a linear chirp that leads to efficient compression or amplification, and thus are particularly useful in the design of optical fiber amplifiers, optical pulse compressors, and solitary wave based communication links. Finally, the stability of the self-similar solutions is discussed numerically under finite initial perturbations.

Phase dynamics of inhomogeneous Manakov vector solitons

We report the exact phase dynamics of Manakov bright and dark vector solitons in an inhomogeneous optical system by means of a variable coefficient coupled nonlinear Schrödinger equation. To investigate the phase dynamics, we have modified the Manakov system with a relation between two modes of propagation, that are obtained by the Hirota bilinear method. The importance of the phase study in soliton interaction is revealed by asymptotic analysis of two-soliton solutions. In contrast with the Manakov bright soliton, the time-dependent dark vector soliton exhibits a gradual phase shift due to the blackness factor. The various inhomogeneous effects on the soliton phase are investigated, with a particular emphasis on nonlinear tunneling. The intensity and corresponding phase of the tunneling soliton either forms a peak or valley and retains its shape after tunneling. Unlike the bright counterpart, the gain or loss term significantly affects the phase of the dark soliton. Apart from the study of soliton intensity, the phase profile of bright and dark vector solitons and its dynamical features are also explored. As the study is not limited to intensity description, the present study could serve as a reference for the future studies on multisolitons phase dynamics in photonics and related fields.

Cubic-quintic nonlinear Helmholtz equation: Modulational instability, chirped elliptic and solitary waves

We study the formation and propagation of chirped elliptic and solitary waves in the cubic-quintic nonlinear Helmholtz equation. This system describes nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities along with spatial dispersion originating from the nonparaxial effect that becomes dominant when the conventional slowly varying envelope approximation fails. We first carry out the modulational instability (MI) analysis of a plane wave in this system by employing the linear stability analysis and investigate the influence of different physical parameters on the MI gain spectra. In particular, we show that the nonparaxial parameter suppresses the conventional MI gain spectrum and also leads to a nontrivial monotonic increase in the gain spectrum near the tails of the conventional MI band, a qualitatively distinct behavior from the standard nonlinear Schrödinger system. We then study the MI dynamics by direct numerical simulations, which demonstrate the production of ultrashort nonparaxial pulse trains with internal oscillations and slight distortions at the wings. Following the MI dynamics, we obtain exact elliptic and solitary wave solutions using the integration method by considering physically interesting chirped traveling wave ansatz. In particular, we show that the system features intriguing chirped antidark, bright, gray, and dark solitary waves depending upon the nature of nonlinearities. We also show that the chirping is inversely proportional to the intensity of the optical wave. In particular, the bright and dark solitary waves exhibit unusual chirping behavior, which will have applications in the nonlinear pulse compression process.

High-power pulse, pulse pair, and pulse train generated by breathers in dispersion exponentially decreasing fiber

Based on the derived rational solutions of the nonautonomous nonlinear Schrödinger equation with varying coefficients, we present a simple scheme to generate a high-power pulse, pulse pair, and pulse train with non-oscillating amplitudes in dispersion exponentially decreasing fiber. Without requiring elimination of the background, the stable pulse train can be generated from the first-order Akhmediev breather, and the high-power pulse and pulse pair can be generated from the second-order Kuznetsov-Ma breather. Moreover, it is found that the characteristics of these pulses can be controlled by adjusting the eigenvalue parameter and fiber parameters. The results presented here are expected to be useful in large-capacity and high-power optical communication systems.

Comprehensive analysis of passive generation of parabolic similaritons in tapered hydrogenated amorphous silicon photonic wires

Parabolic pulses have important applications in both basic and applied sciences, such as high power optical amplification, optical communications, all-optical signal processing, etc. The generation of parabolic similaritons in tapered hydrogenated amorphous silicon photonic wires at telecom (λ ~ 1550 nm) and mid-IR (λ ≥ 2100 nm) wavelengths is demonstrated and analyzed. The self-similar theory of parabolic pulse generation in passive waveguides with increasing nonlinearity is presented. A generalized nonlinear Schrödinger equation is used to describe the coupled dynamics of optical field in the tapered hydrogenated amorphous silicon photonic wires with either decreasing dispersion or increasing nonlinearity. The impacts of length dependent higher-order effects, linear and nonlinear losses including two-photon absorption, and photon-generated free carriers, on the pulse evolutions are characterized. Numerical simulations show that initial Gaussian pulses will evolve into the parabolic pulses in the waveguide taper designed.

Dynamics of vector dark solitons propagation and tunneling effect in the variable coefficient coupled nonlinear Schrödinger equation

We investigate the dynamics of vector dark solitons propagation using variable coefficient coupled nonlinear Schrödinger (Vc-CNLS) equation. The dark soliton propagation and evolution dynamics in the inhomogeneous system are studied analytically by employing the Hirota bilinear method. It is apparent from our asymptotic analysis that the collision between the dark solitons is elastic in nature. The various inhomogeneous effects on the evolution and interaction between dark solitons are explored, with a particular emphasis on nonlinear tunneling. It is found that the tunneling of the soliton depends on a condition related to the height of the barrier and the amplitude of the soliton. The intensity of the tunneling soliton either forms a peak or a valley, thus retaining its shape after tunneling. For the case of exponential background, the soliton tends to compress after tunneling through the barrier/well. Thus, a comprehensive study of dark soliton pulse evolution and propagation dynamics in Vc-CNLS equation is presented in the paper.

Dark soliton pair of ultracold Fermi gases for a generalized Gross-Pitaevskii equation model

We present the theoretical investigation of dark soliton pair solutions for one-dimensional as well as three-dimensional generalized Gross-Pitaevskii equation (GGPE) which models the ultracold Fermi gas during Bardeen-Cooper-Schrieffer-Bose-Einstein condensates crossover. Without introducing any integrability constraint and via the self-similar approach, the three-dimensional solution of GGPE is derived based on the one-dimensional dark soliton pair solution, which is obtained through a modified F-expansion method combined with a coupled modulus-phase transformation technique. We discovered the oscillatory behavior of the dark soliton pair from the theoretical results obtained for the three-dimensional case. The calculated period agrees very well with the corresponding reported experimental result [Weller et al., Phys. Rev. Lett. 101, 130401 (2008)PRLTAO0031-900710.1103/PhysRevLett.101.130401], demonstrating the applicability of the theoretical treatment presented in this work.

Controllable pulse width of bright similaritons in a tapered graded index diffraction decreasing waveguide

We obtain the bright similariton solutions for generalized inhomogeneous nonlinear Schrödinger equation (GINLSE) which governs the pulse propagation in a tapered graded index diffraction decreasing waveguide (DDW). The exact solutions have been worked out by employing similarity transformations which involve the mapping of the GINLSE to standard NLSE for the certain conditions of the parameters. By making use of the exact analytical solutions, we have investigated the dynamical behavior of optical similariton pairs and have suggested the methods to control them as they propagate through DDW. Moreover, pulse width of similariton is controlled through various profiles. These results are helpful to understand the similaritons in DDW and can be potentially useful for future experiments in optical communications which involve optical amplifiers and long-haul telecommunication networks.

Dynamics of kink, antikink, bright, generalized Jacobi elliptic function solutions of matter-wave condensates with time-dependent two- and three-body interactions

By using the F-expansion method associated with four auxiliary equations, i.e., the Bernoulli equation, the Riccati equation, the Lenard equation, and the hyperbolic equation, we present exact explicit solutions describing the dynamics of matter-wave condensates with time-varying two- and three-body nonlinearities. Condensates are trapped in a harmonic potential and they exchange atoms with the thermal cloud. These solutions include the generalized Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. In addition, we have also found rational function solutions. Solutions constructed here have many free parameters that can be used to manipulate and control some important features of the condensate, such as the position, width, velocity, acceleration, and homogeneous phase. The stability of the solutions is confirmed by their long-time numerical behavior.

Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrödinger equation

By using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrödinger equation (NLSE) with cubic and quintic space and time modulated nonlinearities and in the presence of time-dependent and inhomogeneous external potentials and amplification or absorption (source or drain) coefficients. We obtain a class of wide localized exact solutions of NLSE in the presence of a number of non-Hermitian parity-time (PT)-symmetric external potentials, which are constituted by a mixing of external potentials and source or drain terms. The exact solutions found here can be applied to theoretical studies of ultrashort pulse propagation in optical fibers with focusing and defocusing nonlinearities. We show that, even in the presence of gain or loss terms, stable solutions can be found and that the PT symmetry is an important feature to guarantee the conservation of the average energy of the system.

Highly coherent supercontinuum generation with picosecond pulses by using self-similar compression

The low coherence of the supercontinuum (SC) generated using picosecond pump pulses is a major drawback of such SC generation scheme. In this paper, we propose to first self-similarly compress a high power picosecond pump pulse by injecting it into a nonlinearity increasing fiber. The compressed pulse is then injected into a non-zero dispersion-shifted fiber (NZ-DSF) for SC generation. The nonlinearity increasing fiber can be obtained by tapering a large mode area photonic crystal fiber. The fiber nonlinearity is varied by varying the pitch sizes of the air holes. By using the generalized nonlinear Schrödinger equation, we show that a 1 ps pump pulse with random noise can be compressed self-similarly down to a pulse width of 53.6 fs with negligible pedestal. The noise level of the compressed pulse is reduced at the same time. The 53.6 fs pulse can then be used to generate highly coherent SC in an NZ-DSF. By using the proposed scheme, the tolerance of noise level for highly coherent SC generation with picosecond pump pulses can be improved by 5 order of magnitude.

Self-similar propagation and asymptotic optical waves in nonlinear waveguides

The properties of self-similar optical waves propagating in a tapered cubic-quintic nonlinear waveguide are investigated. Using a lens-type transformation we obtain the exact analytical self-similar solutions which describe the propagation of bright-shaped solitons, dark-shaped solitons, kink-shaped solitons, and antikink-shaped solitons. The stability of the solutions is examined by numerical simulations such that stable bright solitons are found. Beyond the exact analytical solutions, asymptotic optical waves are also found by employing a direct ansatz. These waves possess linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves is demonstrated. The analytical results are confirmed by numerical simulations. Finally, we investigate the generation and propagation properties of self-similar optical waves in a quintic nonlinear medium.

Optical rogue waves for the inhomogeneous generalized nonlinear Schrödinger equation

We present optical rogue wave solutions for a generalized nonlinear Schrodinger equation by using similarity transformation. We have predicted the propagation of rogue waves through a nonlinear optical fiber for three cases: (i) dispersion increasing (decreasing) fiber, (ii) periodic dispersion parameter, and (iii) hyperbolic dispersion parameter. We found that the rogue waves and their interactions can be tuned by properly choosing the parameters. We expect that our results can be used to realize improved signal transmission through optical rogue waves.

Binding energy of soliton molecules in time-dependent harmonic potential and nonlinear interaction

We calculate the binding energy of soliton molecules of an integrable nonlinear Schro[over ̈]dinger equation with time-dependent harmonic potential and cubic nonlinearity. Through a scaling transformation, an exact formula for the binding energy can be derived from that of the free soliton molecules in a homogeneous background. In the special case of oscillatory time dependence, sharp resonances occur at some integer and fractional multiples of the natural frequency of the molecule. Enhanced binding is obtained at these resonances and over some finite continuous range of low frequencies.

Matter-wave solutions in Bose-Einstein condensates with harmonic and Gaussian potentials

We study exact matter-wave solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the space- and/or time-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on the similarity transformation and symbolic analysis, we report several families of exact solutions of the quasi-one-dimensional GP equation in the combination of the harmonic and Gaussian potentials, in which some physically relevant solutions are described. The stability of the obtained matter-wave solutions is addressed numerically such that some stable solutions are found. Moreover, we also analyze the parameter regimes for the stable solutions. These results may raise the possibility of relative experiments and potential applications.

Vortex solitons in defocusing media with spatially inhomogeneous nonlinearity

The analytical two- and three-dimensional vortex solitons with arbitrary values of vorticity are constructed in the cubic defocusing media with spatially inhomogeneous nonlinearity. The values of the nonlinearity coefficients are zero near the center and increase rapidly toward the periphery. In addition to the analytical ones, a number of vortex solitons are found numerically. It is shown that analytical vortex solitons are stable. Also, the stability region of the numerically constructed vortex solitons are given.

Amplifier similaritons in a dispersion-mapped fiber laser [Invited]

Amplifier similaritons are generated in a dispersion-mapped fiber laser. Output pulse parameters are nearly independent of the net group velocity dispersion (GVD) owing to the strong local nonlinear attraction in the gain fiber, which dictates the pulse evolution. This constitutes a stable mode-locking regime that is capable of generating sub-100-fs pulses over a broad range of anomalous and normal GVD. These features are consistent with numerical simulations.

Dynamics of a nonautonomous soliton in a generalized nonlinear Schrödinger equation

We solve a generalized nonautonomous nonlinear Schrödinger equation analytically by performing the Darboux transformation. The precise expressions of the soliton's width, peak, and the trajectory of its wave center are investigated analytically, which symbolize the dynamic behavior of a nonautonomous soliton. These expressions can be conveniently and effectively applied to the management of soliton in many fields.

Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation

We derive analytical 3D spatiotemporal similaritons and solitons of the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients.

Soliton tunneling in the nonlinear Schrödinger equation with variable coefficients and an external harmonic potential

We report on the nonlinear tunneling effects of spatial solitons of the generalized nonlinear Schrödinger equation with distributed coefficients in an external harmonic potential. By using the homogeneous balance principle and the F-expansion technique we find the spatial bright and dark soliton solutions. We then display tunneling effects of such solutions occurring under special conditions; specifically when the spatial solitons pass unchanged through the potential barriers and wells affected by special choices of the diffraction and/or the nonlinearity coefficients. Our results show that the solitons display tunneling effects not only when passing through the nonlinear potential barriers or wells but also when passing through the diffractive barriers or wells. During tunneling the solitons may also undergo a controllable compression.

Spatiotemporal nonlinear optical self-similarity in three dimensions

We introduce spatiotemporally expanding self-similar light bullets and vortex torus solutions to the (3+1)D nonlinear Schrödinger equation with gain. In the absence of an initial vorticity, we demonstrate an expanding solution with a parabolic intensity profile and linear spatiotemporal chirp. With a nonzero initial vorticity, expanding vortex torus solutions with a centrally embedded phase singularity are found. Such expanding self-similar structures suggest a route towards a new regime of collapse-free spatiotemporal nonlinear optics.

Propagation of dark similaritons on the compact parabolic background in dispersion-managed optical fibers

The propagation of optical pulses inside dispersion-managed fibers is considered. It is found that the chirped compact parabolic pulse can propagate inside the dispersion-managed fibers self-similarly. Such a finite-width pulse can be served as the background for the propagation and interaction of dark similaritons. Approximate but highly accurate analytical methods are proposed to describe the interaction dynamics of multiple dark similaritons on the self-similar compact parabolic background.

Engineering integrable nonautonomous nonlinear Schrödinger equations

We investigate Painlevé integrability of a generalized nonautonomous one-dimensional nonlinear Schrödinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painlevé analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painlevé integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.

Integrable nonautonomous nonlinear Schrödinger equations are equivalent to the standard autonomous equation

A class of nonautonomous nonlinear Schrödinger equations, claiming to be novel integrable systems with rich properties, continues to appear in the literature. All such equations are shown to be not new, but equivalent to the standard autonomous equation, which trivially explains their integrability features.

Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation

We obtain exact spatiotemporal periodic traveling wave solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients. We utilize these solutions to construct analytical light bullet soliton solutions of nonlinear optics.

Exact solutions of a generalized nonlinear Schrödinger equation

Exact chirped soliton solutions of a generalized nonlinear Schrödinger equation with the cubic-quintic nonlinearities as well as the self-steeping were obtained using a variable parametric method. It was found that the formation of solutions is determined by the sign of a joint parameter solely. By performing numerical simulations, the chirped solutions are stable under perturbations.

Optical similaritons in nonlinear waveguides

We discover analytically an extensive family of optical similaritons, propagating inside graded-index nonlinear waveguide amplifiers. We show that there exists a one-to-one correspondence between these novel similaritons and standard solitons of the homogeneous nonlinear Schrödinger equation. We demonstrate that for certain inhomogeneity and gain profiles, the newly discovered similaritons turn into solitons over sufficiently long propagation distances.

Phase fluctuations of linearly chirped solitons in a noisy optical fiber channel with varying dispersion, nonlinearity, and gain

The phase fluctuations of arbitrarily nonlinearity- and dispersion-managed solitons propagating in a noisy fiber channel are studied both analytically and numerically. We begin by discussing the stability problem of such linearly chirped solitons with a full linear stability analysis. It is shown that these sophisticated solitons possess an enhanced stability against perturbations and therefore hold promise for applications in optical telecommunications. We then make an approach to the phase statistics of these solitons, which stems from an inevitable random walk in phase evolutions due to amplified spontaneous emission noise. By using the variational approach together with impulse-response (Green) functions, an elegant closed-form expression for the phase variance is derived based on an unconstrained self-similar soliton ansatz in which the effect of chirp fluctuations has been critically taken into account as well as the dispersive and nonlinear effects. An inspection of the intriguing subtleties of the interplay among these effects reveals that the chirp fluctuations effect does play an important role in the control of nonlinear phase noise via fiber dispersion, independently of whether the input solitons are initially chirped or not. Our analytical result also offers many possibilities of optimally manipulating nonlinear phase noise with engineered fiber parameters that may lead to the steady pulse propagation, broadening, or compression under favorable parametric conditions. Last, we demonstrate our result by several convincible examples and show an excellent agreement between analytical predictions and numerical simulations.

Experimental demonstration of similariton pulse compression in a comblike dispersion-decreasing fiber amplifier

Self-similar propagation of linearly chirped hyperbolic-secant pulses in a comblike decreasing-dispersion fiber amplifier has been observed experimentally for the first time to our knowledge. The scheme takes advantage of an exact solution of the generalized nonlinear Schrödinger equation with distributed coefficients.

Do solitonlike self-similar waves exist in nonlinear optical media?

We show analytically that bright and dark spatial self-similar waves can propagate in graded-index amplifiers exhibiting self-focusing or self-defocusing Kerr nonlinearities. The intensity profiles of the novel waves are identical with those of fundamental bright or dark spatial solitons supported by homogeneous passive waveguides with the same type of nonlinearity. Thus, we reveal a previously unnoticed connection between spatial solitons and self-similar waves. We also suggest that the discovered self-similar waves can be used in a promising scheme for the amplification and focusing of spatial solitons in future all-optical networks.

Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression

By using the concept of a stationary rescaled pulse (SRP), we analyze an adiabatic soliton compression system based on dispersion-decreasing fiber (DDF). We show that a SRP can exist in a DDF with a linearly decreasing dispersion profile and that the SRP resembles a linearly chirped sech2 pulse. According to the analysis, we show numerically that pedestal-free pulse compression is possible by using the SRP.

Class of solitary wave solutions of the one-dimensional Gross-Pitaevskii equation

We present a large family of exact solitary wave solutions of the one-dimensional Gross-Pitaevskii equation, with time-varying scattering length and gain or loss, in both expulsive and regular parabolic confinement regimes. The consistency condition governing the soliton profiles is shown to map onto a linear Schrödinger eigenvalue problem, thereby enabling one to find analytically the effect of a wide variety of temporal variations in the control parameters, which are experimentally realizable. Corresponding to each solvable quantum mechanical system, one can identify a soliton configuration. These include soliton trains in close analogy to experimental observations of Streckeret al. [Nature (London) 417, 150 (2002)], spatiotemporal dynamics, solitons undergoing rapid amplification, collapse and revival of condensates, and analytical expression of two-soliton bound states, to name a few.

Self-similar propagation and compression of chirped self-similar waves in asymmetric twin-core fibers with nonlinear gain

Ultrashort-pulse propagation in asymmetric twin-core fiber amplifiers is studied with the aid of self-similarity analysis of the nonlinear Schrödinger-type equation interacting with a source, variable dispersion, variable Kerr nonlinearity, variable gain or loss, and nonlinear gain. Exact chirped pulses that can propagate self-similarly subject to simple scaling rules of this model have been found. It is reported that the pulse position of these chirped pulses can be precisely piloted by appropriately tailoring the dispersion profile. This fact is profitably exploited to achieve optimal pulse compression of these chirped self-similar solutions.

Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients

The generalized nonlinear Schrödinger model with distributed dispersion, nonlinearity, and gain or loss is considered and the explicit, analytical solutions describing the dynamics of bright solitons on a continuous-wave background are obtained in quadratures. Then, the generation, compression, and propagation of pulse trains are discussed in detail. The numerical results show that solitons can be compressed by choosing the appropriate control fiber system, and pulse trains generated by modulation instability can propagate undistorsted along fibers with distributed parameters by controlling appropriately the energy of each pulse in the pulse train.

Stationary rescaled pulse in alternately concatenated fibers with O1-accumulated nonlinear perturbations

Pulse propagation in a transmission line comprising alternately concatenated fibers with O1-accumulated perturbations of self-phase modulation and anomalous dispersion is studied. In such a line, a pulse compression process is inevitable. In certain conditions, we have found that a compressed pulse can be rescaled to the initial pulse, and hence that there exists a stationary rescaled pulse (SRP), which is distinct from other nonlinear stationary pulses such as the guiding-center, dispersion-managed, or split-step solitons. The properties of SRP are studied. We apply the rescaling to the transmission line rather than to the pulse, and we experimentally observe a SRP in a periodically rescaled transmission line.

Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity

Three novel types of self-similar solutions, termed parabolic, Hermite-Gaussian, and hybrid pulses, of the generalized nonlinear Schrödinger equation with varying dispersion, nonlinearity, and gain or absorption are obtained. The properties of the self-similar evolutions in various nonlinear media are confirmed by numerical simulations. Despite the diversity of their formations, these self-similar pulses exhibit many universal features which can facilitate significantly the achievement of well-defined linearly chirped output pulses from an optical fiber, an amplifier, or an absorption medium, under certain parametric conditions. The other intrinsic characteristics of each type of self-similar pulses are also discussed.

Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients

A broad class of exact self-similar solutions to the nonlinear Schrödinger equation (NLSE) with distributed dispersion, nonlinearity, and gain or loss has been found describing both periodic and solitary waves. Appropriate solitary wave solutions applying to propagation in optical fibers and optical fiber amplifiers with these distributed parameters have also been studied in detail. These solutions exist for physically realistic dispersion and nonlinearity profiles. They correspond either to compressing or spreading solitary pulses which maintain a linear chirp or to chirped oscillatory solutions. The stability of these solutions has been confirmed by numerical simulations of the NLSE with perturbed initial conditions. These self-similar propagation regimes are expected to find practical application in both optical fiber amplifier systems and in fiber compressors.

Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrodinger equation

We consider the inhomogeneous higher-order nonlinear Schrodinger equation and explicitly present exact combined solitary wave solutions that can describe the simultaneous propagation of bright and dark solitary waves in a combined form in inhomogeneous fiber media or in optical communication links with distributed parameters. Furthermore, we analyze the features of the solutions, and numerically discuss the stabilities of these solitary waves under slight violations of the parameter conditions and finite initial perturbations. The results show that there exist combined solitary wave solutions in an inhomogeneous fiber system, and the combined solitary wave solutions are stable under slight violations of the parameter conditions and finite initial perturbations. Finally, the interaction between two neighboring combined solitary waves is numerically discussed.

Nonlinear compression of solitary waves in asymmetric twin-core fibers

We demonstrate a different pulse compression technique based on exact solutions to the nonlinear Schrödinger-type equation interacting with a source, variable dispersion, variable Kerr nonlinearity, and variable gain or loss. We show that this model is appropriate for the pulse propagation in asymmetric twin-core fibers. The chirped pulses are compressed due to the nonlinearity as well as dispersion management as also due to the space dependence of the gain coefficient. We also obtain singular solitary wave solutions, pertaining to extreme increase of the amplitude due to self-focusing.

Chirped self-similar solutions of a generalized nonlinear Schrödinger equation model

Exact chirped self-similar solutions of the generalized nonlinear Schrödinger equation with varying dispersion, nonlinearity, gain or absorption, and nonlinear gain have been found. The stability of these nonlinearly chirped solutions is then demonstrated numerically by adding Gaussian white noise and by evolving from an initial chirped Gaussian pulse, respectively. It is reported that the pulse position of these chirped pulses can be precisely piloted by tailoring the dispersion profile, and that the sech-shaped solitary waves can propagate stably in the regime of beta(z)gamma(z) > 0 as well as the regime of beta(z)gamma(z) < 0 , according to the magnitude of the nonlinear chirp parameter. Our theoretical predictions are in excellent agreement with the numerical simulations.