NLS Model Research Articles

Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons

The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schr\"odinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sine-Gordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential $V =\eta I^2 - \frac{\epsilon}{6} I^3 $ and the saturable type potential satisfying $V'[I] =2 \eta I - \frac{\epsilon I^q}{1+ I^q},\,q \in \IZ_{+}$, with a deformation parameter $\epsilon \in \IR$ and $I=|\psi|^2$. The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of $\{\eta, \epsilon, n\}$. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.

Modulational instability and nano-scale energy localization in ferromagnetic spin chain with higher order dispersive interactions

The nonlinear localization phenomena in ferromagnetic spin lattices have attracted a steadily growing interest and their existence has been predicted in a wide range of physical settings. We investigate the onset of modulational instability of a plane wave in a discrete ferromagnetic spin chain with physically significant higher order dispersive octupole–dipole and dipole–dipole interactions. We derive the discrete nonlinear equation of motion with the aid of Holstein–Primakoff (H–P) transformation combined with Glauber's coherent state representation. We show that the discrete ferromagnetic spin dynamics is governed by an entirely new discrete NLS model with complex coefficients not reported so far. We report the study of modulational instability (MI) of the ferromagnetic chain with long range dispersive interactions both analytically in the frame work of linear stability analysis and numerically by means of molecular dynamics (MD) simulations. Our numerical simulations explore that the analytical predictions correctly describe the onset of instability. It is found that the presence of the various exchange and dispersive higher order interactions systematically favors the local gathering of excitations and thus supports the growth of high amplitude, long-lived discrete breather (DB) excitations. We analytically compute the strongly localized odd and even modes. Further, we employ the Jacobi elliptic function method to solve the nonlinear evolution equation and an exact propagating bubble-soliton solution is explored.

SVAR and NLS Model Analysis of US Bank Financial Derivatives and the Related Fields Based on Their Statistical Data

SVAR and NLS Model Analysis of US Bank Financial Derivatives and the Related Fields Based on Their Statistical Data

Role of economic factors in terrorism in Pakistan

This research paper analyzed the role of economic factors in terrorism in Pakistan empirically using the annual time series data, covering the period from 2001 to 2014. The stationarity of the variables was checked by applying the augmented Dicky–Fuller unit root test. The NLS and ARMA (least square regression) model have been used as analytical techniques. The results revealed that except poverty all other economic factors (unemployment, income inequality, GDP per capita, literacy rate, population density and inflation rate) included in the study show the positive and significant impact on terrorism in case of Pakistan. In the recommendations the study suggests that economic factors plays role in terrorism in Pakistan. Government should need to control these factors by giving possible and satisfactory solution.

Modelling of the spatial evolution of extreme laboratory wave Heights with the nonlinear Schrödinger and Dysthe equations

The spatial evolution of nonlinear wave groups with the same initial Gaussian envelope are theoretically studied under different kind of governing equations, confirming that the higher order nonlinear Schrödinger equation formulation of Dysthe is capable of representing the asymmetric features of propagating wave packets which have narrow spectral bandwidth in deep water. Moreover, laboratory experiments run in an offshore wave basin are systematically compared with a large set of numerical simulations, respectively performed by using the third-order and fourth-order nonlinear wave models, mainly focusing on the aspects of higher order statistical moments such as the coefficients of skewness and kurtosis of the surface elevation, and the exceedance distributions of wave heights in different random sea states. The modulational instability, resulting from a quasi-resonant four wave interaction in the unidirectional sea state, can be indicated by the coefficient of kurtosis, which has shown an appreciable correlation with the abnormal wave density from a statistical point of view. Compared with Dysthe simulation, the NLS model presents a more intensive energy focusing, particularly at the intermediate stage of evolution process in the presence of strong nonlinearity.

Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model

We present a collective coordinate approximation to model the dynamics of two interacting nonlinear Schrödinger solitons. We discuss the accuracy of this approximation by comparing our results with those of the full numerical simulations and find that the approximation is remarkably accurate when the solitons are some distance apart, and quite reasonable also during their interaction.

Hydrodynamic Supercontinuum

We report the experimental observation of multi-bound-soliton solutions of the nonlinear Schrödinger equation (NLS) in the context of hydrodynamic surface gravity waves. Higher-order N-soliton solutions with N=2, 3 are studied in detail and shown to be associated with self-focusing in the wave group dynamics and the generation of a steep localized carrier wave underneath the group envelope. We also show that for larger input soliton numbers, the wave group experiences irreversible spectral broadening, which we refer to as a hydrodynamic supercontinuum by analogy with optics. This process is shown to be associated with the fission of the initial multisoliton into individual fundamental solitons due to higher-order nonlinear perturbations to the NLS. Numerical simulations using an extended NLS model described by the modified nonlinear Schrödinger equation, show excellent agreement with experiment and highlight the universal role that higher-order nonlinear perturbations to the NLS play in supercontinuum generation.

On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions

We consider a general class of discrete nonlinear Schroedinger equations (DNLS) on the lattice $h \mathbb{Z}$ with mesh size $h>0$. In the continuum limit when $h \to 0$, we prove that the limiting dynamics are given by a nonlinear Schroedinger equation (NLS) on $\mathbb{R}$ with the fractional Laplacian $(-\Delta)^\alpha$ as dispersive symbol. In particular, we obtain that fractional powers $1/2 < \alpha < 1$ arise from long-range lattice interactions when passing to the continuum limit, whereas NLS with the non-fractional Laplacian $-\Delta$ describes the dispersion in the continuum limit for short-range lattice interactions (e.g., nearest-neighbor interactions). Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.

The concept of quasi-integrability for modified non-linear Schrödinger models

We consider modifications of the nonlinear Schrodinger model (NLS) to look at the recently introduced concept of quasi-integrability. We show that such models possess an infinite number of quasi-conserved charges which present intriguing properties in relation to very specific space-time parity transformations. For the case of two-soliton solutions where the fields are eigenstates of this parity, those charges are asymptotically conserved in the scattering process of the solitons. Even though the charges vary in time their values in the far past and the far future are the same. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. Our findings may have important consequences on the applications of these models in several areas of non-linear science. We make a detailed numerical study of the modified NLS potential of the form V = |psi|^(2(2+epsilon)), with epsilon being a perturbation parameter. We perform numerical simulations of the scattering of solitons for this model and find a good agreement with the results predicted by the analytical considerations. Our paper shows that the quasi-integrability concepts recently proposed in the context of modifications of the sine-Gordon model remain valid for perturbations of the NLS model.

SELECTED TOPICS IN CLASSICAL INTEGRABILITY

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integrals of motion and the corresponding Lax pair are extracted based on algebraic considerations. Our attention is restricted to classical discrete and continuum integrable systems with periodic boundary conditions. Typical examples of discrete (Toda chain, discrete NLS model) and continuum integrable models (NLS, sine–Gordon models and affine Toda field theories) are also discussed.

Families of quasi-rational solutions of the NLS equation and multi-rogue waves

We construct a multi-parametric family of the solutions of the focusing nonlinear Schrödinger equation (NLS) from the known results describing the multi-phase almost-periodic elementary solutions given in terms of Riemann theta functions. We give a new representation of their solutions in terms of Wronskians determinants of order 2N composed of elementary trigonometric functions. When we perform a special passage to the limit when all the periods tend to infinity, we obtain a family of quasi-rational solutions. This leads to efficient representations for the Peregrine breathers of orders N = 1, 2, 3 first constructed by Akhmediev and his co-workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue-wave formation in the frame of the NLS model first explained by V B Matveev in 2010. Our formulation allows us to isolate easily the second- or third-order Peregrine breathers from ‘generic’ solutions and also to compute the Peregrine breathers of orders 2 and 3 easily with respect to other approaches. In the cases N = 2, 3, we obtain the comfortable formulas to study the deformation of a higher Peregrine breather of order 2 to the three rogue-wave or order 3 to the six rogue-wave solutions via the variation of the free parameters of our construction.

Polarization Reversal Kinetics in Strontium Barium Niobate Relaxor Crystals

The domain nucleation and growth during the switching process in ferroelectric relaxor Sr0.61Ba0.39Nb2O6 single crystals have been observed by means of nematic liquid crystal (NLC) method. Microscale studies of switching process in conjunction with electrical measurements allowed us to establish a relationship between local properties of domain dynamics and macroscopic response i.e., polarization hysteresis loop and switching currents. For the interpretation of experimental data we utilized two alternative theories concerning the domain growth: the Kolmogorov-Avrami-Ishibashi (KAI model) that involves nucleation and unrestricted growth of domains and nucleation—limited switching (NLS model) focused on the statistics of nucleation.

Continuous approximation of breathers in one- and two-dimensional DNLS lattices

In this paper we construct and approximate breathers in the DNLS model starting from the continuous limit: such periodic solutions are obtained as perturbations of the ground state of the NLS model in , with n = 1, 2. In both the dimensions we recover the Sievers–Takeno and the Page (P) modes; furthermore, in also the two hybrid (H) modes are constructed. The proof is based on the interpolation of the lattice using the finite element method (FEM).

Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams

We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that collapse in the simpler NLS model. Thereby, we provide the first ever numerical evidence that nonparaxiality and backscattering can arrest the collapse. We also solve the (1+1)D NLH and show that solitons with radius of only half the wavelength can propagate over forty diffraction lengths with no distortions. In both cases we calculate the backscattered field, which has not been done previously. Finally, we compute the dynamics of counter-propagating solitons using the NLH model, which is more comprehensive than the previously used coupled NLS model.

Q-Boson in Quantum Integrable Systems

q-bosonic realization of the underlying Yang-Baxter algebra is identified for a series of quantum integrable systems, including some new models like two-mode q-bosonic model leading to a coupled two-component derivative NLS model, wide range of q-deformed matter-radiation models, q-anyon model etc. Result on a new exactly solvable interacting anyon gas, linked to q-anyons on the lattice is reported.

QUANTUM SOLITONS

Two themes are considered in this paper. First of all in the Introduction a comment is made about the Tzitzéica equation which occurred in the author's work as reported in [1]. Much recent work has referred to this equation and a representative bibliography is given in this paper with brief comments. The second theme of this paper is concerned with work surrounding Quantum Solitons and the author's talk at the ISLAND 2 meeting under the title ‘Quantum solitons’ is briefly summarised. An extended review of this subject matter has now appeared in [2] and as there the quantum soliton of the quantum attractive NLS model is seen as a ‘qubit’ for quantum information purposes. It is hoped that this summary and/or the reference [2] can help to stimulate further interest in this ‘quantum’ aspect of our subject in the solitons community. The actual observation of a quantum soliton is also reported and a proper theoretical description of it given. Reference is made to -boson lattices first as a simple example of the quantum inverse method in the Introduction and then as a subject matter in its own right at the end of the paper.

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.

Off shell dynamics of the O(3) NLS model beyond Monte Carlo and perturbation theory

The off-shell dynamics of the O(3) non-linear sigma model is probed in terms of spectral densities and two-point functions by means of the form factor approach. The exact form factors of the spin field, Noether current, EM tensor and the topological charge density are computed up to six particles. The corresponding n ⩽ 6 particle spectral densities are used to compute the two-point functions, and are argued to deviate at most a few per mille from the exact answer in the entire energy range below 103 in units of the mass gap. To cover yet higher energies we propose an extrapolation scheme to arbitrary particle numbers based on a novel scaling hypothesis for the spectral densities. It yields candidate results for the exact two-point functions at all energy scales and allows us to exactly determine the values of two, previously unknown, non-perturbative constants.

Effects of nonlocal dispersive interactions on self-trapping excitations

A one-dimensional discrete nonlinear Schr\odinger (NLS) model with the power dependence ${\mathrm{r}}^{\mathrm{\ensuremath{-}}\mathrm{s}}$ on the distance r of the dispersive interactions is proposed. The stationary states ${\mathrm{\ensuremath{\psi}}}_{\mathrm{n}}$ of the system are studied both analytically and numerically. Two types of stationary states are investigated: on-site and intersite states. It is shown that for s sufficiently large all features of the model are qualitatively the same as in the NLS model with a nearest-neighbor interaction. For s less than some critical value ${\mathrm{s}}_{\mathrm{cr}}$, there is an interval of bistability where two stable stationary states exist at each excitation number N=${\ensuremath{\sum}}_{\mathrm{n}}$|${\mathrm{\ensuremath{\psi}}}_{\mathrm{n}}$${\mathrm{|}}^{2}$. For cubic nonlinearity the bistability of on-site solitons may occur for dipole-dipole dispersive interaction (s=3), while ${\mathrm{s}}_{\mathrm{cr}}$ for intersite solitons is close to 2.1. For increasing degree of nonlinearity \ensuremath{\sigma}, ${\mathrm{s}}_{\mathrm{cr}}$ increases. The long-distance behavior of the intrinsically localized states depends on s. For sg3 their tails are exponential, while for 23 they are algebraic. In the continuum limit the model is described by a nonlocal NLS equation for which the stability criterion for the ground state is shown to be s+1.

Perturbative versus non-perturbative QFT lessons from the O(3) NLS model

The two-point functions of the energy-momentum tensor and the Noether current are used to probe the O(3) nonlinear sigma model in an energy range below 10 4 in units of the mass gap m. We argue that the form factor approach, with the form factor series truncated at the 6-particle level, provides an almost exact solution of the model in this energy range. The onset of the (2-loop) perturbative regime is found to occur only at energies around 100 m.