Nonlinear Schrodinger Equation Model Research Articles

Some Models for the Interaction of Long and Short Waves in Dispersive Media: Part I—Derivation

It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Korteweg–de Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves, and such a system has been studied over the last few decades. However, questions about the validity of this system in the study of water waves were raised in a previous work of one of us where the analysis was presented using the fifth-order KdV as the starting point. These questions will now be settled unequivocally in a series of papers. In this first part, we show that the NLS-KdV system (or even the linear Schrodinger–KdV system) cannot be resulted from the full Euler equations formulated in the study of water waves. In the process of so doing, we also propose a few alternative models for describing the interaction of long and short waves.

Study on characteristics of internal solitary waves in the Malacca Strait based on Sentinel-1 and GF-3 satellite SAR data

The study of the characteristics of internal solitary waves happened in the Malacca Strait is an urgent problem for submarine, ship navigation and marine engineering. Based on SAR remote sensing data obtained from the high spatial resolution Sentinel-1 and GF-3, the internal solitary wave characteristics of the Malacca Strait are investigated. By use of 20 Sentinel-1 SAR images from June 2015 to December 2016 and 24 GF-3 images from April 2018 to March 2019, the spatial distribution characteristics of internal solitary wave s are statistically analyzed. It is found that the internal solitary waves are usually in the form of wave packets and single solitary waves, and the maximum crest length of the leading wave can reach 39 km. The amplitude and group velocity of internal solitary wave s can be calculated by the inversion model of high-order nonlinear Schrodinger (NLS) equation, and the calculated amplitude of the internal solitary wave s and the propagation group velocity of the wave packets range from 4.7 m to 23.9 m and 0.12 m/s to 0.40 m/s, respectively. The range of phase velocity of single internal solitary waves obtained by KdV equation is 0.26 m/s to 0.60 m/s. In general, the amplitude and the velocity of internal solitary wave s in Malacca strait are related to the topography.

Linearly chirped mid-infrared supercontinuum in all-normal-dispersion chalcogenide photonic crystal fibers.

We demonstrate all-normal dispersion supercontinuum generation in chalcogenide photonic crystal fibers pumped at 2070-2080 nm with a femtosecond fiber laser. At 2.9 kW peak power, the generated supercontinuum has a 3 dB bandwidth of 27.6 THz and -20 dB bandwidth of 75.5 THz. We experimentally investigated the supercontinuum evolution inside our sample fiber at various peak powers and fiber lengths and study the impact of fiber water absorption on the generated supercontinuum spectrum. Multiple tests with fiber length- ranging from 0.34 to 10 cm-provide insight on pulse evolution along fiber length. Our simulations, which utilizes the generalized nonlinear Schrodinger equation model, match perfectly the experiments for all tested pump powers and fiber lengths, and confirm that the output pulse has a linear chirp, allowing linear pulse compression.

On the Design of NFT-Based Communication Systems With Lumped Amplification

Nonlinear Fourier transform (NFT) based transmission technique relies on the integrability of the nonlinear Schrodinger equation (NLSE). However, the lossless NLSE is not directly applicable for the description of light evolution in fibre links with lumped amplification such as Erbium-doped fibre amplifier (EDFA) because of the nonuniform loss and gain evolution. In this case, the path-averaged model is usually applied as an approximation of the true NLSE model including the fibre loss. However, the inaccuracy of the lossless path-average model, even though being small, can also result in a notable performance degradation in NFT-based transmission systems. In this paper, we extend the theoretical approach, which was first proposed for solitons in EDFA systems, to the case of NFT-based systems to constructively diminish the aforementioned performance penalty. Based on the quantitative analysis of distortions due to the use of path-average model, we optimise the signal launch and detection points to minimise the models mismatch. Without loss of generality, we demonstrate how this approach works for the NFT systems that use continuous NFT spectrum modulation (vanishing signals) and NFT main spectrum modulation (periodic signals). Through numerical modelling, we quantify the corresponding improvements in system performance.

Traveling wave solutions of the nonlinear Schrödinger equation

In this paper, we investigate the traveling soliton and the periodic wave solutions of the nonlinear Schrödinger equation (NLSE) with generalized nonlinear functionality. We also explore the underlying close connection between the well-known KdV equation and the NLSE. It is remarked that both one-dimensional KdV and NLSE models share the same pseudoenergy spectrum. We also derive the traveling wave solutions for two cases of weakly nonlinear mathematical models, namely, the Helmholtz and the Duffing oscillators' potentials. It is found that these models only allow gray-type NLSE solitary propagations. It is also found that the pseudofrequency ratio for the Helmholtz potential between the nonlinear periodic carrier and the modulated sinusoidal waves is always in the range 0.5 ≤ Ω/ω ≤ 0.537285 regardless of the potential parameter values. The values of Ω/ω = {0.5, 0.537285} correspond to the cnoidal waves modulus of m = {0, 1} for soliton and sinusoidal limits and m = 0.5, respectively. Moreover, the current NLSE model is extended to fully NLSE (FNLSE) situation for Sagdeev oscillator pseudopotential which can be derived using a closed set of hydrodynamic fluid equations with a fully integrable Hamiltonian system. The generalized quasi-three-dimensional traveling wave solution is also derived. The current simple hydrodynamic plasma model may also be generalized to two dimensions and other complex situations including different charged species and cases with magnetic or gravitational field effects.

Nonlinear interaction between surface plasmons and ion oscillations in a semi-bounded collisional quantum plasma

In this paper, we have investigated the nonlinear interaction between high-frequency surface plasmons and low-frequency ion oscillations in a semi-bounded collisional quantum plasma. By coupling the nonlinear Schrodinger equation and quantum hydrodynamic model, and taking into account the ponderomotive force, the dispersion equation is obtained. By solving this equation, it is shown that there is a modulational instability in the system, and collisions and quantum forces play significant roles on this instability. The quantum tunneling increases the phase and group velocities of the modulated waves and collisions increase the growth rate of the modulational instability. It is also shown that the effect of quantum forces and collisions is more significant in high modulated wavenumber regions.

Temporal dynamics of optical bistability and modulation instability in colloidal nanoparticles

Colloidal nanoparticles are attractive optical materials for their low threshold nonlinear response and thus all-optical, nondestructive features which are important in biomedical optics and optical processing. We develop a theoretical scheme based on numerical solution of Nonlinear Schrodinger Equation and nonideal gas model of nonlinearity to investigate temporal analysis of optical bistability (OB) and modulation instability in colloidal nanoparticles. Our scheme determines the dependence of a nanosuspension system dynamic state on characteristic/control parameters including external feedback depth, nanosuspension length, and the initial density of nanoparticles as well as the optical input power. We show that these parameters are intensely correlated. We also indicate that the nonlinear response of nanosuspension may be saturated over a threshold of input power, and thus an unexpected procedure of system evolution toward stability rather than transition to chaos will occur. Consequently, provided that internal feedback is present inside the nanosuspension controlling chaos will be attainable by simply adjusting the optical input power as the control parameter in contrast to the other chaos control methods which require external injection. Finally, we propose an approach which gives a measure of switching time to optimize OB. The optimum results are obtained for the lowest taken values of characteristic/control parameters.

Characteristics of filaments at high-power femtosecond laser radiation propagation in air and water: II. Numerical simulation

The results of numerical solution of the nonlinear Schrodinger equation modeling nonstationary self-focusing and filamentation of high-power ultra-short laser radiation in atmospheric air and water are presented. The comparison of the major characteristics of the light and plasma filaments formed in these media is carried out.

The nonlinear evolution of rogue waves generated by means of wave focusing technique

Generating the rogue waves in offshore engineering is investigated, first of all, to forecast its occurrence to protect the offshore structure from being attacked, to study the mechanism and hydrodynamic properties of rouge wave experimentally as well as the rouge/structure interaction for the structure design. To achieve these purposes demands an accurate wave generation and calculation. In this paper, we establish a spatial domain model of fourth order nonlinear Schrodinger (NLS) equation for describing deep-water wave trains in the moving coordinate system. In order to generate rogue waves in the experimental tank efficiently, we take care that the transient water wave (TWW) determines precisely the concentration of time/place. First we simulate the three-dimensional wave using TWW in the numerical tank and modeling the deepwater basin with a double-side multi-segmented wave-maker in Shanghai Jiao Tong University (SJTU) under the linear superposing theory. To discuss its nonlinearity for guiding the experiment, we set the TWW as the initial condition of the NLS equation. The differences between the linear and nonlinear simulations are presented. Meanwhile, the characteristics of the transient water wave, including water particle velocity and wave slope, are investigated, which are important factors in safeguarding the offshore structures.

Self-action of tightly focused femtosecond laser radiation in air in a filamentation regime: Laboratory and numerical experiments

The influence of self-focusing of strong femtosecond laser radiation on the spatial energy distribution of a laser beam was studied. Experimental data are provided on the transverse energy density distribution of the tightly focused radiation of the Ti-sapphire laser following its filamentation in air. The results are interpreted within the framework of the nonlinear Schrodinger equation model. The best agreement between the theory and the experiment is found in the model that takes into account the change of character of the nonlinear response of the medium induced by high-intensity radiation in the nonlinear focus, which includes the change of the ionization mechanism of molecules from multiphoton ionization to tunnel ionization, the completely instantaneous Kerr nonlinearity, and saturation of the latter due to higher-order nonlinearity. The mechanism of the ionization of molecules can change from purely multiphoton ionization to tunnel ionization.

The Maxwell-Lorentz model for optical pulses

Dynamics of optical pulses, especially of ultra short femtosecond pulses, are of great technological and theoretical interest. The dynamics of optical pulses is usually studied using the nonlinear Schrodinger (NLS) equation model. While such approach works surprisingly well for description of pulse propagation, at least in the femtosecond regime, the full system posses a wealth of new wave phenomena that we explore in this paper: envelope collapse regularization resulting in the orignal pulse splitting; development of infinite gradients in the carrier wave; existence of the stable top hat traveling wave solutions formed by a pair of kink anti-kink shaped optical waves.

Periodic Optical Pulses in Nonlinear Optical Fibers

We discuss the problem of transmitting polarized pulses along optical fibers with variable dispersion. The dissipation and mean dispersion are assumed to be zero, which allows using the model of the vector nonlinear Schrodinger equation. We consider an optical fiber consisting of arms of equal length, which is assumed to be large. We propose an asymptotic recursive procedure for calculating the amplitude and the phase of an optical pulse propagating along the optical cable with variable dispersion.

The inverse problem of chirp pulse propagating through Ti:sapphire amplifier

Summary By using the nonlinear Schrodinger equation model, the inverse problem of the chirp pulse through the Ti:sapphire amplifier was studied, in which the effects of gain narrowing, gain saturation and amplified spontaneous emission on the pulse were considered. The results were useful for the design of the pulse shaping of the ultrashort pulse.

Characteristics of supercontinuum generationin tapered fibers using femtosecond laser pulses

Drawing single mode fibers over a flame generates tapered fibers with waist diameters of approximately 1–3 micrometers and waist lengths of up to 90 mm. We demonstrate how the profile of such tapered fibers can be determined. We then characterize the white light that is generated in a variety of such fibers, showing its dependence on waist length and waist diameter and demonstrating its dependence on pulse parameters such as pulse duration, spectral position, and pulse power. A comparison with theoretical calculations using a nonlinear Schrodinger equation model including Kerr nonlinearities is given. Furthermore, we show XFROG spectrograms of the pulses propagating through tapered fibers, confirming the model of soliton splitting in the anomalous dispersion regime.

Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers

Supercontinuum generation in a highly nonlinear, dispersion-shifted fiber at 1550 nm is discussed. Spectrum generation under both pulsed and continuous-wave conditions is considered. With a few meters of highly nonlinear, dispersion-shifted fiber and a femtosecond erbium fiber laser, an octave-spanning supercontinuum is demonstrated. Kilometer lengths of nonlinear fiber pumped by a continuous-wave Raman fiber laser are shown to generate a continuum with a bandwidth greater than 247 nm. A nonlinear Schrodinger-equation model is used to investigate the effect of varying the dispersion on the pulsed continuum and noise effects in the continuous-wave continuum.

Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves

The critical nonlinear Schrodinger equation (NLS) models the propagation of intense laser light in Kerr media. This equation is derived from the more comprehensive nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. It is known that if the input power of the laser beam (i.e., L2 norm of the initial solution) is sufficiently high, then the NLS model predicts that the beam will self-focus to a point (i.e., collapse) at a finite propagation distance. Mathematically, this behavior corresponds to the formation of a singularity in the solution of the NLS. A key question which has been open for many years is whether the solution to the NLH, i.e., the parent equation, may nonetheless exist and remain regular everywhere, particularly for those initial conditions (input powers) that lead to blowup in the NLS. In the current study we address this question by introducing linear damping into both models and subsequently comparing the numerical solutions of the damped NLH (boundary-value problem) with the corresponding solutions of the damped NLS (initial-value problem) for the case of one transverse dimension. Linear damping is introduced in much the same way as is done when analyzing the classical constant-coefficient Helmholtz equation using the limiting absorption principle. Numerically, we have found that it provides a very efficient tool for controlling the solutions of both the NLH and NLS. In particular, we have been able to identify initial conditions for which the NLS solution does become singular, while the NLH solution still remains regular everywhere. We believe that our finding of a larger domain of existence for the NLH than for the NLS is accounted for by precisely those mechanisms that have been neglected when deriving the NLS from the NLH, i.e., nonparaxiality and backscattering.

Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion

We show that the methodology based on the generalized inverse scattering transform (IST) concept provides a systematic way to discover the novel exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain or absorption. The fundamental innovation of the present approach is to notice that it is possible both to allow for a variable spectral parameter with new dependent variables and to apply of the famous moving in time focuses concept of the self-focusing theory to the IST formalism. We show that for nonlinear optics this algorithm is a useful tool to design novel dispersion managed fiber transmission lines and soliton lasers. Fundamental soliton management regimes are predicted.

Novel soliton solutions of the nonlinear Schrodinger equation model

The methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark "soliton islands" in a "sea of solitary waves" of the nonlinear Schrodinger equation model with varying dispersion, nonlinearity, and gain or absorption. It is shown that solitons exist only under certain conditions and the parameter functions describing dispersion, nonlinearity, and gain or absorption inhomogeneities cannot be chosen independently. Fundamental soliton management regimes are discovered.

Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain

The novel stable “soliton islands” in a “sea of solitary waves” of the nonlinear Schrodinger equation model with varying dispersion, nonlinearity, and gain or absorption are discovered. Different soliton management regimes are predicted.

Structure of zero modes in a model of the discrete (2+1)-dimensional nonlinear Schrödinger equation

The structure of the zero modes in a discrete (2+1)-dimensional model of the gauge-invariant nonlinear Schrodinger equation is studied. Including the compactification of the Chern-Simons gauge fields eliminates the difficulties with the continuous model [L. A. Abramyan and A. P. Protogenov, JETP Lett. 64, 859 (1996); L. A. Abramyan, V. I. Berezhiani, and A. P. Protogenov, Phys. Rev. E 56, 6026 (1997)] and leads to a prediction of the existence of a transition region characterized by a hierarchical sequence of collapses which are enumerated by the Chern-Simons coefficient. Using the zero modes in calculating the dependence of the critical power N on the Chern-Simons coefficient, we have found that the transition region lies in the interval 11.703≤N≤12.01.