Nonlinear Schrodinger Equation Research Articles

Existence and nonexistence of global solutions for time-dependent damped NLS equations

Existence and nonexistence of global solutions for time-dependent damped NLS equations

Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \mathrm{i}\partial_t q(x, t)+\partial_{x}^2q(x, t)+2\sigma q^{2}(x, t)\overline{q(-x, t)} = 0 $ with initial data $ q(x, 0)\in H^{1, 1}(\mathbb{R}) $. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $ H^{1, 1} $ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

DYNAMICAL EVOLUTION STUDY FOR ONE-DIMENSIONAL SYSTEM WITH CUBIC-QUINTIC NONLINEARITY

Nonlinear Schrodinger equation (NLSE) is a quantum system of nonlinear evolution， a kind of important equations, and in many areas have very important applications. Therefore, this type of equation models has important physical significance. People realized with GGPE describing BCS-BEC crossover, GGPE began to be utilized in people's research. When GPE and polytropic approximation are combined, the nonlinear term becomes generalized cubic-quintic nonlinearity. And the model becomes one-dimensional generalized cubic-quintic nonlinear Schrodinger equation (GCQ-NLSE). Many prior work deal with the special circumstances GCQ-NLSE the CQ-NLSE, and need to introduce additional integrability condition. In this work, we use the phase and amplitude coupling Transform combined with F-expansion method to solve the one-dimensional GCQ-NLSE. We obtained its soliton solutions without introducing any integrable condition. We apply the analytical solution obtained to our quantitative analysis of dynamic behavior of the model, whose results are dependent on the polytropic index which is related to the intensity of soliton strength, speed, system phonon velocity, these are depicted pictorially.

Method of Line Solution of Optical Soliton in Optical Communication System

The performance of fiber-optic communication is constrained by the dispersive phenomenon of group velocity dispersion (GVD) and fiber non-linearity of self-phase modulation, but when these effects are balanced, a single wave or soliton is produced. A soliton can travel a great distance with constant speed. It is suitable for use in ultra-long-distance communication because of these unique features. The non-linear Schrodinger (NLS) equation describes the optical soliton propagation in optical fibers. By using the method of line (MOL) to solve the NLS equation, this study mimics the optical soliton propagation. It demonstrates that during simulations at 400 km, the optical soliton maintains its shape and amplitude.

Soliton formation and dynamics in the quintic nonlinear media with [formula omitted]-invariant harmonic-Gaussian potential

In this paper, we introduce a PT-symmetric harmonic-hyperbolic-Gaussian (HHG) potential in the nonlinear Schrödinger (NLS) equation with quintic nonlinearity. Firstly, we present the critical thresholdless PT-phase symmetry breaking in the linear regime. Secondly, we give both exact and numerical single-/double-hump solitons of the quintic NLS equations with the PT-symmetric HHG potential, and show their stabilities by studying the spectrum stability analysis and direct wave propagations. Finally, we investigate the interactions between the single-/double-hump solitons and the exotic hyperbolic secant function, as well as the adiabatic excitations of the solitons with the operation of an adiabatic switch function. All these obtained results provide a theoretical guidance for the applications of the PT-symmetric theory into diverse areas of physics.

A supersymmetric generalized NLS equation: linear spectral problems and beyond

A supersymmetric generalized NLS equation: linear spectral problems and beyond

Gaussian solitary solution for a class of logarithmic nonlinear Schrödinger equation in (1 + n) dimensions

The Gaussian solitary wave solutions are obtained for a class of logarithmic nonlinear Schrödinger equation (NLSE) with a standard harmonic oscillator potential. We find that the Gaussian solution does not exist at all in the straight line x=vt, but it does in the curve x=vt. We illustrate the behavior of the Gaussons and analyze the correlations with respect to the relevant parameters. The logarithmic nonlinearity as an imaginary term enables us to get the associated advection equation, thus showing in a novel way the existence of a Gaussian solutions in the curve x=vt. Moreover, we generalize the present study to the high-dimensional (1 + n) NLSE case. These results will enrich the current literature on this nonlinear equation.

Mitigating the Inter-Core Crosstalk of Multicore Fiber Transmission by Orthogonal Filtering

We for the first time proposed to utilize the orthogonal filtering technique to suppress the inter-core crosstalk (IC-XT) in multicore fiber (MCF) transmission. It could simplify the implementation of weakly-coupled MCF (WC-MCF) signal reception without multiple-input and multiple-output (MIMO) digital signal processing, by extracting the individual lanes separately. Effective IC-XT mitigation using a pair of orthogonal filters are demonstrated for MIMO-free 2-core MCF transmitted signal receptions. 600-Gbps 64-QAM (quadrature amplitude modulation) signal transmission over 100-km MCF is emulated by constructing a concrete transmission model based on coupled nonlinear Schrodinger equations. Maximum Q factor improvement of 6 dB can be achieved using orthogonal matching filters to suppress IC-XT at −10dB. The effect of spatial mode dispersion on the filter orthogonality performance is also analyzed.

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the ‘mass’ (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick’s theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick’s theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the ‘frustrated’ blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

Study of optical solitons for Kudryashov’s Quintuple power-law with dual form of nonlinearity using two modified techniques

The main purpose of this article is to explore the analytic soliton solutions of a nonlinear Schrodinger equation (NLSE), including Kudryashova’s quintuple power-law with dual form nonlinearity by presenting two efficient analytical techniques. The present equation is a significant model which deals with a variety of nonlinear phenomenons in the fields of optical fibers. Modified G′G2 -expansion technique and exp(−φ(ρ)) technique explore kink-singular, bright, dark and cupson-singular form of soliton solutions. The present techniques are simple and efficient because these techniques produced a rational, trigonometric solution of the nonlinear partial differential equation. The above two techniques are novel and have not been applied yet on present model.

From the AKNS system to the matrix Schrödinger equation with vanishing potentials: Direct and inverse problems

AbstractWe relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schrödinger equation. The corresponding Miura transformation, which allows this connection, converts the focusing matrix nonlinear Schrödinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.

Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves

We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schrodinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such solutions has only been established against co-periodic perturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbations which are localized, i.e., integrable on the line. Such localized perturbations naturally yield the absence of a spectral gap, so we must rely on a substantially different method with origins in the stability analysis of periodic waves in reaction-diffusion systems. The relevant linear estimates have been obtained in recent work by the first three authors through a delicate decomposition of the associated linearized solution operator. Since its most critical part just decays diffusively, the nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation. However, the modulated perturbation satisfies a quasilinear equation yielding an inherent loss of regularity which, due to the low-order damping in the LLE, cannot be regained using standard methods. Therefore, in contrast to the case of reaction-diffusion systems, the nonlinear iteration cannot be closed for the modulated perturbation. In this work, we present a new nonlinear iteration scheme incorporating tame estimates on the unmodulated perturbation, which satisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to close an independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLE against localized perturbations with precisely the same decay rates as predicted by the linear theory.

A nonlinear Schrödinger equation including the parabolic law and its dark solitons

The main goal of the current paper is to analyze nonlinear effects on dynamical feathers of soliton waves in a nonlinear Schrödinger equation (NLSE) including the parabolic law. To this end, after applying the complex envelope and distinguishing real and imaginary structures, a group of dark solitons to the governing model are retrieved using the Kudryashov method. Based on the results, by adjusting the values of nonlinear effects, the amplitude of the dark solitons can be easily controlled.

Bound-state soliton gas as a limit of adiabatically growing integrable turbulence

We study numerically the integrable turbulence in the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type using a new approach called the “growing of turbulence”. In this approach, we add a small linear pumping term to the equation and start evolution from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and examine the resulting integrable turbulence. For sufficiently small initial noise and pumping coefficient, and also for not very wide simulation box (basin length), we observe that the turbulence grows in a universal adiabatic regime, moving successively through the statistically stationary states of the integrable 1D-NLSE, which do not depend on the pumping coefficient, amplitude of the initial noise or basing length. Waiting longer in the growth stage, we transit from weakly nonlinear states to strongly nonlinear ones, characterized by a high frequency of rogue waves. Using the inverse scattering transform (IST) method to monitor the evolution, we observe that the solitonic part of the wavefield becomes dominant even when the (linear) dispersion effects are still leading in the dynamics and with increasing average intensity the wavefield approaches a dense bound-state soliton gas, whose properties are defined by the Fourier spectrum of the initial noise. Regimes deviating from the universal adiabatic growth also lead to solitonic states, but solitons in these states have noticeably different velocities and a significantly wider distribution by amplitude, while the statistics of wavefield indicates a much more frequent appearance of very large waves.

An Approximate Solution for the Non-Linear Fractional Schrodinger Equation with Harmonic Oscillator

An Approximate Solution for the Non-Linear Fractional Schrodinger Equation with Harmonic Oscillator

Darboux transformation and exact solutions of NLS-MB equations

A nonlinear wave is one of the basic objects of physics. They are inherent to plasma physics and solid state physics, gravity and nuclear physics, field theory and optics, hydrodynamics and aerodynamics, kinetics of chemical reactions and population dynamics. It is well known that the construction of explicit solutions for an integrable system plays a significant part in the definition and explanation of nonlinear phenomena. In this article, we will focus on integrable nonlinear Schrodinger and Maxwell-Bloch equations (NLS-MB) that represents the propagation of optical impulses in an inhomogeneous fiberglass with erbium-doped losses or amplification due to an external potential. Lax representation of NLS-MB will be given. Based on relevant Lax pair, Darboux transformation for NLS-MB will be obtained. Exact solutions will be derived through the Darboux transformation. Graphs of the obtained solutions will be constructed. By using our approach one can find also other differerent exact solutions of NLS-MB equations.

A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform

The increasing demand for bandwidth and long-haul transmission has led to new methods of signal processing and transmission in optical fiber communication systems. The nonlinear Fourier transform is one of the most recent methods proposed, and is able to represent an integrable nonlinear Schrödinger equation (NLSE) channel in terms of its continuous and discrete spectrum, to overcome the limitation of the bandwidth imposed by the Kerr effect on silica fibers. In this paper, we will propose and investigate the Boffetta-Osburne method for the direct nonlinear Fourier implementation, and the Gel’fand-Levitan-Marchenko equation for the inverse nonlinear Fourier, as only the continuous part of the nonlinear spectrum will be used to encode information. A novel methodology is proposed to improve their numerical implementation with respect to the NLSE, and we analyze in detail how the improved algorithm can be used in a real optical system, by investigating three different modulation schemes. We report increased performance transmission and consistency in the numerical results when the proposed methodology is applied. Our results show that b-modulation will increase the Q-factor by 2 dB with respect to the other two modulations. The improvement results with our proposed methodology suggest that b-modulation applied only to a continuous part of the nonlinear spectrum is a very effective method for maximizing both the transmission bandwidth and distance in optical fiber communication systems.

Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation

We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrödinger equation with small potential and the nonlinear Schrödinger equation (NLSE) with weak nonlinearity. For the Schrödinger equation with small potential characterized by a dimensionless parameter ε ∈ ( 0 , 1 ] \varepsilon \in (0, 1] , we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in L 2 L^2 -norm to prove a uniform error bound at time t ε = t / ε t_\varepsilon =t/\varepsilon as C ( t ) C ~ ( T ) ( h m + τ 2 ) C(t)\widetilde {C}(T)(h^m +\tau ^2) up to t ε ≤ T ε = T / ε t_\varepsilon \leq T_\varepsilon = T/\varepsilon for any T &gt; 0 T&gt;0 and uniformly for ε ∈ ( 0 , 1 ] \varepsilon \in (0,1] , while h h is the mesh size, τ \tau is the time step, m ≥ 2 m \ge 2 and C ~ ( T ) \tilde {C}(T) (the local error bound) depend on the regularity of the exact solution, and C ( t ) = C 0 + C 1 t C(t) =C_0+C_1t grows at most linearly with respect to t t with C 0 C_0 and C 1 C_1 two positive constants independent of T T , ε \varepsilon , h h and τ \tau . Then by introducing a new technique of regularity compensation oscillation (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform (w.r.t ε \varepsilon ) error bound at O ( h m − 1 + ε τ 2 ) O(h^{m-1} + \varepsilon \tau ^2) is established in H 1 H^1 -norm for the long-time dynamics up to the time at O ( 1 / ε ) O(1/\varepsilon ) of the Schrödinger equation with O ( ε ) O(\varepsilon ) -potential with m ≥ 3 m \geq 3 . Moreover, the RCO technique is extended to prove an improved uniform error bound at O ( h m − 1 + ε 2 τ 2 ) O(h^{m-1} + \varepsilon ^2\tau ^2) in H 1 H^1 -norm for the long-time dynamics up to the time at O ( 1 / ε 2 ) O(1/\varepsilon ^2) of the cubic NLSE with O ( ε 2 ) O(\varepsilon ^2) -nonlinearity strength. Extensions to the first-order and fourth-order time-splitting methods are discussed. Numerical results are reported to validate our error estimates and to demonstrate that they are sharp.

Variational analysis of averaged nonlinearity-managed solitons

The nonlinear Schrodinger equation with periodically varying nonlinearity coefficient thatdescribes matter-wave soliton in in Bose-Einstein Condensation (BEC) and optical solitons isconsidered analytically. Variational approach is applied on the derived averaged equationwhich describes nonlinearity-managed solitons. A good agreement between analytical solutions obtained by means of variational analysis and numerical solutions of derived averagedequation is demonstrated.

Optical Properties of Photonic Crystal Fibers with Fluid Cores

The objective of this research is to compare pulses travelling through different photonic crystal fiber cores (DPCFC) using both finite element method (FEM) /split step Fourier method (SSFM). A unique DCPCF design with exceptionally high non-linearity has been introduced to achieve ultra-high pulse amplitude. Via a generalized non-linear equation, we evaluate the refractive index ratio and dispersion, for each type which is used, as well as output amplitude for different cores in different photonic crystal fiber core by utilizing the solution for the nonlinear Schrodinger equation (NLSE). Lastly, the findings are compared to different photonic crystal fiber design parameters. This paper provides a photonic crystal fiber design consisting of multiple liquid cores and theoretically solved non-linear equations. It was discovered in this design that the refractive index of propylene glycol (C3H8O2) is greater than ethylene glycol’s (C2H6O2), and that both are far greater than silica's refractive index (SiO2). Propylene (C3H8O2) has a lower dispersion than ethylene glycol (C2H6O2) and silica (SiO2). The output amplitudes of (C3H8O2) and (C2H6O2) were then shown to be substantially bigger than the output amplitudes of (SiO2) with respect to distance and time. This emphasizes the need for using certain liquids as cores in holey fibers, dependent upon their use.