Diffraction Decreasing System Research Articles

High-dimensional vector solitons for a variable-coefficient partially nonlocal coupled Gross–Pitaevskii equation in a harmonic potential

A (3+1)-dimensional variable-coefficient partially nonlocal coupled Gross–Pitaevskii equation trapped in a harmonic potential becomes a focus of this paper. A counterpart of this variable-coefficient coupled equation is found as a (2+1)-dimensional constant-coefficient single nonlinear Schrödinger equation via the reduction procedure. By solutions of constant-coefficient single equation via the Hirota method, and from this counterpart, analytical high-dimensional vector soliton solutions with the Hermite–Gaussian envelope of the variable-coefficient coupled equation are deduced. Expanded behaviors of high-dimensional vector solitons emerge in the exponential diffraction decreasing system.

Excitation manipulation of three-dimensional completely localized rogue waves in a partially nonlocal and inhomogeneous nonlinear medium

We study a (2 + 1)-dimensional variable-coefficient-coupled partially nonlocal nonlinear Schrodinger equation with nonlinearities localized in x and y-directions and non-localized in z-direction, and set up a one-to-one connection between it and the standard nonlinear Schrodinger equation. Using this one-to-one connection, and with the help of bilinear method, we analytically derive first-order and second-order rogue wave solutions including rogue wave triplet solution, which are completely localized in three-dimensional space. By modulating the maximal value of the effective propagation distance, and comparing this maximal value with the top (peak) position of rogue wave excitation in the exponential diffraction decreasing system, we discuss the excitation manipulation of three-dimensional rogue waves, including original excitation, top excitation, tail excitation and fast complete excitation of single rogue wave and rogue wave triplet.

Management of Controllable Two-Kuznetsov–Ma Soliton in 𝒫𝒯-Symmetric Dimers

Abstract We study a (2+1)-dimensional coupled nonlinear Schrödinger equation in 𝒫𝒯-symmetric inhomogeneous nonlinear dimers with different values of diffractions and derive a two-Kuznetsov–Ma soliton solution with 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry based on the Darboux and similarity transformation methods. In a diffraction decreasing system with exponential profile, we investigate the management of controllable two-Kuznetsov–Ma soliton by modulating the relation between the values of the maximal effective propagation distance Z m and periodic peak locations Z ij . If Z m <Z ij , Z m =Z ij and Z m >Z ij , we discuss the initial excitation, peak excitation, and complete excitation of the structure with the i th part and j th peak for two-Kuznetsov–Ma soliton.

Recurrence behavior for controllable excitation of rogue waves in a two-dimensional $$\varvec{\mathcal {PT}}$$ PT -symmetric coupler

A (2 + 1)-dimensional variable-coefficient coupled nonlinear Schrodinger equation with different diffractions in parity-time symmetric coupler is studied, and exact solutions in the form of two-component Peregrine solution and rogue wave triplet are derived. Based on these solutions, by adjusting the relation between the maximal value \(Z_\mathrm{m}\) and the exciting location values \(Z_0\) for Peregrine solution and \(Z_1,Z_2\) for rogue wave triplet, recurrence behaviors for controllable excitation of Peregrine solution and rogue wave triplet including complete excitation, peak excitation, rear excitation and initial excitation are discussed in the exponential diffraction decreasing system. This phenomenon of recurrence for controllable excitation is owing to different values of diffractions in two transverse directions.

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.

Repetitive excitations of controllable vector breathers in a nonlinear system with different distributed transverse diffractions from arterial mechanics and optical fibers

A (2+1)-dimensional coupled nonlinear Schrodinger equation with different distributed transverse diffractions is studied, and vector breather solutions are analytically obtained. By adjusting the relation between the maximum effective distance \(Z_m\) and the effective distance, \(Z_j\), with the peak amplitude, the control for the evolutional behaviors of the vector breather is discussed. Moreover, controllable vector breathers exhibit the phenomenon of repetitive excitations in a diffraction decreasing system, which originates from different values of transverse diffractions.

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 localized solitons in cubic and power-law competing nonlinear media with different diffractions and $$\mathcal {PT}$$ PT -symmetric potentials

Analytical spatiotemporal localized soliton solutions in cubic and power-law competing nonlinear media with different diffractions and \(\mathcal {PT}\)-symmetric potentials are derived. The compression and expansion of localized soliton structures in the diffraction decreasing system and periodic modulation system are studied analytically and numerically. The numerical simulation shows that exact solutions exist stable region in the defocusing cubic and focusing quintic nonlinear medium. However, exact solutions in the focusing cubic and defocusing quintic nonlinear medium are unstable, namely they cannot maintain their original shapes, they are distorted and collapse, and they ultimately decay into noise. These results might stimulate the related experiment to observe localized structures of nonlinear Schrodinger equation in \(\mathcal {PT}\)-symmetric systems.

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.

Vector combined and crossing Kuznetsov–Ma solitons in $$\mathcal {PT}$$ PT -symmetric coupled waveguides

A variable-coefficient coupled nonlinear Schrodinger equation with balanced gain and loss is studied, and two kinds of analytical Kuznetsov–Ma soliton solutions including combined Kuznetsov–Ma soliton solution and crossing two-Kuznetsov–Ma soliton solution are obtained. In a diffraction decreasing system, the control for the excitation of two kinds of Kuznetsov–Ma soliton including rear excitation, peak excitation and initial excitation is investigated. The different peak locations of Kuznetsov–Ma soliton along the propagation direction appear repeatedly, which makes rear excitation, peak excitation and initial excitation happen again and again.

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.

Hollow ring-like soliton and dipole soliton in ( $$\varvec{2+1}$$ 2 + 1 )-dimensional $$\varvec{\mathcal {PT}}$$ PT -symmetric nonlinear couplers with gain and loss

By means of the Hirota bilinear method, we obtain the hollow ring-like soliton and dipole soliton solutions of the coupled nonlinear Schrodinger equation in the ( $$2+1$$ )-dimensional inhomogeneous $${\mathcal {PT}}$$ -symmetric nonlinear couplers. Based on these analytical solutions, we investigate the compression behaviors of hollow ring-like soliton and dipole soliton in a diffraction decreasing system with exponential profile.

Vortex soliton in (2+1)-dimensional $$\mathcal {PT}$$ PT -symmetric nonlinear couplers with gain and loss

We study the coupled nonlinear Schrodinger equation in the (2+1)-dimensional inhomogeneous \(\mathcal {PT}\)-symmetric nonlinear couplers and obtain \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric vortex soliton solutions. The dynamical behaviors of the completely localized structures (vortex solitons) are analytically and numerically investigated in a diffraction decreasing system with exponential profile. Numerical results indicate that one vortex soliton with different topological charges can stably propagate a long distance. The space between two humps and the modulation depth of vortex solitons add when the topological charge increases. However, the change tendency of the amplitude and width of vortex solitons is same with the increase in topological charge.

Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in $${\varvec{\mathcal {PT}}}$$ PT -symmetric nonlinear couplers with gain and loss

We investigate a (2+1)-dimensional-coupled variable coefficient nonlinear Schrodinger equation in parity time symmetric nonlinear couplers with gain and loss and analytically obtain a combined structure solution via the Darboux transformation method. When the imaginary part of the eigenvalue \(n\) is smaller or bigger than 1, we can obtain the combined Peregrine soliton and Akhmediev breather, or Kuznetsov–Ma soliton, respectively. Moreover, we study the controllable behaviors of this combined Peregrine soliton and Kuznetsov–Ma soliton structure in a diffraction decreasing system with exponential profile. In this system, the effective propagation distance \(Z\) exists a maximal value \(Z_m\) and the maximum amplitude of the KM soliton appears in the periodic locations \(Z_{i}\). By modulating the relation between values of \(Z_m\) and \(Z_i\), we realize the control for the excitation of the combined Peregrine soliton and Kuznetsov–Ma soliton, such as the restraint, maintenance, and postpone.

Controllable Akhmediev breather and Kuznetsov-Ma soliton trains in PT-symmetric coupled waveguides.

The PT-symmetric and PT-antisymmetric Akhmediev breather (AB) and Kuznetsov-Ma (KM) soliton train solutions of a (2+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equation in PT-symmetric coupled waveguides with gain and loss are derived via the Darboux transformation method. From these analytical solutions, we investigate the controllable behaviors of AB and KM soliton trains in a diffraction decreasing system with exponential profile. By adjusting the relation between the maximum Zm of effective propagation distance and the peak locations Zi of AB and KM soliton trains, we can control the restraint, maintenance and postpone excitations of AB and KM soliton trains.

Gaussian-type light bullet solutions of the ([formula omitted])-dimensional Schrödinger equation with cubic and power-law nonlinearities in [formula omitted]-symmetric potentials

Two kinds of Gaussian-type light bullet (LB) solutions of the (3+1)-dimensional Schrödinger equation with cubic and power-law nonlinearities in PT-symmetric potentials are analytically obtained. The phase switches, powers and transverse power-flow densities of these solutions in homogeneous media are studied. The linear stability analysis of these LB solutions and the direct numerical simulation indicate that LB solutions are stable below some thresholds for the imaginary part of PT-symmetric potentials in the defocusing cubic and focusing power-law nonlinear medium, while they are always unstable for all parameters in other media. Moreover, the broadened and compressed behaviors of LBs in the exponential periodic amplification system and diffraction decreasing system are discussed. Results indicate that LB is more stable for the sign-changing nonlinearity in the exponential periodic amplification system than for the non-sign-changing nonlinearity in the diffraction decreasing system at the same propagation distances.