Nonlocal Nonlinear Schrodinger Equation Research Articles

Vector spatiotemporal localized structures in (3 $$+$$ + 1)-dimensional strongly nonlocal nonlinear media

We investigate the (3 \(+\) 1)-dimensional coupled nonlocal nonlinear Schrodinger equation in the inhomogeneous nonlocal nonlinear media and derive analytical vector spatiotemporal localized solution. Based on this solution, Gaussian solitons and some symmetric multipole patterns around the point \((x, y) =(0,0)\) can be constructed. The change trends of the amplitude and width of solitons are opposite, and they finally tend to a certain value. The compression and expansion of spatiotemporal localized structures are also studied in an exponential diffraction decreasing system.

Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential.

The integrable nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential [M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013)] is investigated, which is an integrable extension of the standard nonlinear Schrödinger equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation (1,N-1)-fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical behaviors of these higher-order rational solitons with some small noises on the basis of numerical simulations.

Soliton solutions for the nonlocal nonlinear Schrödinger equation

The Darboux transformation (DT) for the integrable nonlocal nonlinear Schrodinger equation (nNLSE) is constructed with the aid of loop group method. Based on the DT, we derive several types of analytical solutions for the focusing case and defocusing case, such as the periodical solutions, soliton solutions, breathers and rational ones. Dynamical properties for those solutions to the nNLSE are analyzed on basis of the solution expressions and figures.

Hermite–Gaussian vortex solitons of a (3+1)-dimensional partially nonlocal nonlinear Schrödinger equation with variable coefficients

We consider the wave motion in a partially nonlocal and inhomogeneous nonlinear medium, and a (3+1)-dimensional nonlocal nonlinear Schrodinger equation with variable coefficients is used to govern this dynamics. Based on this model, spatiotemporal Hermite–Gaussian vortex soliton solutions are derived. The evolution behaviors of spatiotemporal Hermite–Gaussian vortex solitons in a diffraction decreasing system are investigated. Results indicate that the topological charge m changes the spiral structures of phase, and its value determines the number of the branch of the spiral phase structures. If the value of parameter n adds, spatiotemporal vortex solitons change their structures. Obviously, the layer of ring solitons along the vertical (z-axis) direction is decided by \(n+1\).

Spatiotemporal Hermite–Bessel solitons in (3+1)-dimensional strongly nonlocal nonlinear $$\varvec{\mathcal {PT}}$$ PT -symmetric media

The propagation of optical solitons in the coupled dual waveguides with balanced gain and loss and nonlocal nonlinearity is governed by the coupled nonlocal nonlinear Schrodinger equation with \(\mathcal {PT}\)-symmetry. We study this model and obtain analytical spatiotemporal Hermite–Bessel soliton solutions. The control of expansion and compression of spatiotemporal Hermite–Bessel soliton in different axis directions are studied in the diffraction decreasing system. Moreover, the periodic compression and expansion of spatiotemporal Hermite–Bessel solitons in different axis directions are also discussed in the periodic distributed amplification system.

Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation

A nonlocal nonlinear Schrödinger (NLS) equation was recently introduced and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this paper a detailed study of the inverse scattering transform of this nonlocal NLS equation is carried out. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained. The inverse scattering theory is developed by using a novel left–right Riemann–Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg–de Vries and sine-Gordon equations are made.

Self-similar azimuthons in strongly nonlocal nonlinear media with $${\varvec{{\mathcal {PT}}}}$$ PT -symmetry

A (2+1)-dimensional coupled nonlocal nonlinear Schrodinger equation is studied in \({\mathcal {PT}}\)-symmetric nonlocal nonlinear couplers with gain and loss, and approximate self-similar solution of vector rotating azimuthon is obtained. Various structures of rotating azimuthon and soliton cluster are constructed. Moreover, dynamical behaviors of azimuthon and soliton cluster are discussed in the constant diffraction system and the exponential diffraction decreasing waveguide.

Spatiotemporal Hermite–Gaussian solitons of a (3 + 1)-dimensional partially nonlocal nonlinear Schrödinger equation

A (3 + 1)-dimensional partially nonlocal nonlinear Schrodinger equation is considered, and approximate spatiotemporal Hermite–Gaussian soliton solutions are obtained using the Hirota method. Based on these results, some basic characteristics of spatiotemporal Hermite–Gaussian solitons are studied.

Spatiotemporal localizations in $$(3+1)$$ ( 3 + 1 ) -dimensional $${{\mathcal {PT}}}$$ PT -symmetric and strongly nonlocal nonlinear media

A \((3+1)\)-dimensional coupled nonlocal nonlinear Schrodinger equation in the inhomogeneous \({\mathcal {PT}}\)-symmetric and strongly nonlocal nonlinear media is studied, and analytical vector spatiotemporal localized solutions are obtained. From these solutions, Gaussian soliton clusters, multipole soliton clusters, and nested soliton clusters can be constructed. The expansion behavior and periodic expansion and compression of spatiotemporal localizations are also investigated in the diffraction decreasing system and periodic modulation system, respectively.

Vector Hermite–Gaussian spatial solitons in (2+1)-dimensional strongly nonlocal nonlinear media

We obtain an analytical vector Hermite–Gaussian spatial soliton solution of the (2+1)-dimensional coupled nonlocal nonlinear Schrodinger equation in the inhomogeneous nonlocal nonlinear media, and investigate the periodic expansion and compression behaviors of Hermite–Gaussian spatial solitons in a periodic modulation system. The structure of Hermite–Gaussian soliton lattice is decided by the degree (n, m) of Hermite polynomials. The evolution of the soliton-lattice breather appears the full breathing cycle, and the interval between solitons oscillates periodically as the wave propagates. The amplitude and width change periodically; however, they exist opposite trend in the periodic modulation system.

Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrödinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL, and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only the elliptic functions dn(x, m) and cn(x, m) with modulus m but also their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lamé polynomials of order 1 but also admits solutions in terms of Lamé polynomials of order 2, even though they are not the solution of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.

Generalized Darboux Transformation and Rational Solutions for the Nonlocal Nonlinear Schrödinger Equation with the Self-Induced Parity-Time Symmetric Potential

In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.

Exact solitons in three-dimensional weakly nonlocal nonlinear time-modulated parabolic law media

The three-dimensional (3D) nonlocal nonlinear Schrödinger equation (NNLSE) with time-dependent parabolic law nonlinearity and an external potential, describes the propagation of an optical pulse in the weakly nonlocal nonlinear parabolic law media, has been studied analytically. Explicit solutions are constructed by using the Jacobian elliptic equations (JEEs) expansion method. The 3D nonlocal spatial bright and dark optical solitons have been found.

Two-dimensional accessible solitons in PT-symmetric potentials

Two-dimensional parity-time (PT) symmetric potentials are introduced, which allow the existence of spatial solitons in the model of the strongly nonlocal nonlinear Schrodinger equation. Two-dimensional accessible solitons are found in the form of solutions separating the radial amplitude, given in terms of Laguerre polynomials, a phase function involving quadratic, linear, and constant phase shifts, and a specific azimuthal modulation function. Shape-preserving solitons are constructed from Laguerre–Gaussian functions containing the self-similar variable and an exponential form of the azimuthal modulation, containing sine and cosine functions, when a suitable PT-symmetric potential is chosen. Interesting soliton profiles and the corresponding PT-symmetric potentials are displayed for different values of the parameters.

Propagation properties of Airy–Gaussian beams

The propagation of the Airy–Gaussian beams is studied in strongly nonlocal nonlinear media analytically and numerically. The linear momentum of the analytical Airy–Gaussian beam solution of the Snyder–Mitchell model is not conservational, which is the reason that results in the disagreement between the analytical Airy–Gaussian beam solution and the numerical simulations of the nonlocal nonlinear Schrodinger equation in the case of strong nonlocality. The quasi-Airy–Gaussian soliton in the Gaussian-shaped response material can be obtained when the parameter χ0 is large enough, and the patterns of Airy–Gaussian beams are variable periodically in liquid crystal material during propagation.

INVESTIGATION ON PROPAGATION CHARACTERISTICS OF SUPER-GAUSSIAN BEAM IN HIGHLY NONLOCAL MEDIUM

We investigate the propagation characteristics of super- Gaussian beam in highly nonlocal nonlinear media. The optical beam propagation has been modeled by well known nonlocal nonlinear Schrodinger equation. The variational method is employed to flnd the initial beam propagation parameters and then split step Fourier method is used for numerical simulations. A generalized exact analytical solution of the model is obtained and critical power of soliton is determined. The evolution of super-Gaussian beam has shown oscillatory propagation.

On the Cauchy problem for the nonlocal derivative nonlinear Schrödinger equation

We consider the Cauchy problem associated with the one-dimensional nonlocal derivative nonlinear Schrodinger equation, and establish local well-posedness for “small” initial data in the usual L-based Sobolev spaces H(R), s > 1/2. We also prove that our result is “almost sharp” in the sense that the flow-map data-solution fails to be C at the origin from H(R) to H(R) for any s < 1/2. Finally, thanks to the lack of energy conservation, we prove the nonexistence of solitary-wave solutions.

Accessible solitary wave families of the generalized nonlocal nonlinear Schrödinger equation

Two-dimensional accessible solitary wave families of the generalized nonlocal nonlinear Schrodinger equation are obtained by utilizing superpositions of various single accessible solitary solutions. Specific values of soliton parameters are selected as initial conditions and the superposition of known single solitary solutions in the highly nonlocal regime are launched into the nonlocal nonlinear medium with a Gaussian response function, to obtain novel numerical solitary solutions of improved stability. Our results reveal that in nonlocal media with the Gaussian response the higher-order spatial accessible solitary families can exist in various forms, such as asymmetric necklace, asymmetric fractional, and symmetric multipolar necklace solitons.

Solitons and spectral renormalization methods in nonlinear optics

Localized wave solutions, often referred to as solitary waves or solitons, are important classes of solutions in nonlinear optics. In optical communications, weakly nonlinear, quasi-monochromatic waves satisfy the “classical” and the “dispersion-managed” nonlocal nonlinear Schrodinger equations, both of which have localized pulses as special solutions. Recent research has shown that mode-locked lasers are also described by similar equations. These systems are variants of the classical nonlinear Schrodinger equation, appropriately modified to include terms which model gain, loss and spectral filtering that are present in the laser cavity. To study their remarkable properties, a computational method is introduced to find localized waves in nonlinear optical systems governed by these equations.

Propagation of fractional charge Laguerre–Gaussian light beams in moving defocusing media with thermal nonlinearity

Propagation of the Laguerre–Gaussian (LG) light beams with fractional topological charges in a moving defocusing nonlinear medium is studied. The fractional charge LG beam is constructed as a weighted superposition of the standard LG modes with integer charges. Our model is governed by a nonlocal nonlinear Schrodinger equation and describes in particular the beam propagation in the atmosphere along the earth surface with allowance for thermal nonlinearity and in the presence of a cross-wind. The numerical simulations have revealed the remarkable stability of the cross-section intensity distribution in the fractional charge LG beams during their propagation, as compared with the case of integer charge beams.