79 pags., 30 figs., 3 apps. -- Open Access funded by Creative Commons Atribution Licence 3.0

Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems

In this paper, a $$(3+1)$$
 -dimensional nonlinear evolution equation is cast into Hirota bilinear form with a dependent variable transformation. A bilinear Backlund transformation is then presented, which consists of six bilinear equations and involves nine arbitrary parameters. With multiple exponential function method and symbolic computation, nonresonant-typed one-, two-, and three-wave solutions are obtained. Furthermore, two classes of lump solutions to the dimensionally reduced cases with $$y=x$$
 and $$y=z$$
 are both derived. Finally, some figures are given to reveal the propagation of multiple wave solutions and lump solutions.

Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation

(INVITED) Vortex solitons: Old results and new perspectives

Generalized complex Ginzburg–Landau equation (GCGLE) can be used to describe the nonlinear dynamic characteristics of fiber lasers and has riveted much attention of researchers in ultrafast optics. In this paper, analytic solutions of the GCGLE are obtained via the modified Hirota bilinear method. Kink waves and period waves are presented by selecting the relevant parameters. The influence of the related parameters on them is analyzed and studied. The results indicate that the desired pulses can be demonstrated by effectively controlling the dispersion and nonlinearity of fiber lasers.

Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers

Hierarchies of Peregrine solution and breather solution are derived in a (2+1)-dimensional variable-coefficient nonlinear Schrodinger equation with partial nonlocality. Based on these solutions, we study the control of the excitation of Peregrine solution and breather solution in different planes. In particular, the localized Peregrine solution and breather solution are firstly reported in two-dimensional space. It is expected that our analysis and results may give new insight into higher-dimensional localized rogue waves in nonlocal media.

Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality

A (3+1)-dimensional nonlinear Schrodinger equation with variable-coefficient dispersion/diffraction and cubic-quintic-septimal nonlinearities is studied, two families of analytical light bullet solutions with two types of $${{\mathcal {PT}}}$$
 -symmetric potentials are obtained. The coefficient of the septimal nonlinear term strongly influences the form of light bullet. The direct numerical simulation indicates that light bullet solutions in different cubic-quintic-septimal nonlinear media exhibit different property of stability, and under different $${\mathcal {PT}}$$
 -symmetric potentials they also show different stability against white noise. These stabilities of evolution originate from subtle interplay among dispersion, diffraction, nonlinearity and $${\mathcal {PT}}$$
 -symmetric potential. Moreover, compression and expansion of light bullets in the hyperbolic dispersion/diffraction system and periodic modulation system are investigated numerically. The evolution of light bullet in periodic modulation system is more stable than that in the hyperbolic dispersion/diffraction system.

Dynamics of light bullets in inhomogeneous cubic-quintic-septimal nonlinear media with {\varvec{{\mathcal {PT}}}}-symmetric potentials

A (2+1)-dimensional N-coupled nonlinear Schrodinger equation with spatially modulated cubic–quintic nonlinearity and transverse modulation is studied, and vector multipole and vortex soliton solutions are analytically obtained. When the modulation depth q is chosen as 0 and 1, vector multipole and vortex solitons are constructed, respectively. The number of “petals” for the multipole solitons and vortex solitons is related to the value of the topological charge m, and the number of layers in the multipole solitons and vortex solitons is determined by the value of the soliton order number n.

Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity

Gaussian spatial soliton solutions of both the constant-coefficient and variable-coefficient (2 + 1)-dimensional nonlinear Schrodinger equations in quintic–septimal nonlinear materials with different diffractions are presented under two kinds of $${\mathcal {P}}{\mathcal {T}}$$
 -symmetric potentials. The linear stability analysis and direct numerical simulation are jointly utilized to investigate the stability for analytical solutions of the constant-coefficient equation. Results from the linear stability analysis and the direct numerical simulation possess a high degree of consistency, that is, the stable case for Gaussian spatial solitons of the constant-coefficient equation appears only in the defocusing quintic and focusing septimal nonlinear material. Moreover, reconstruction of stable Gaussian spatial solitons of the variable-coefficient equation is studied based on the expression of the effective propagation distance Z(z) by choosing an appropriate form of diffraction $$\beta _1(z)$$
 .

Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under $${{\varvec{\mathcal {P}}}}{\varvec{\mathcal {T}}}$$ P T -symmetric potentials

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.

Yan Fan

Papers

Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality

Dynamics of light bullets in inhomogeneous cubic-quintic-septimal nonlinear media with {\varvec{{\mathcal {PT}}}}-symmetric potentials

Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity

Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under $${{\varvec{\mathcal {P}}}}{\varvec{\mathcal {T}}}$$ P T -symmetric potentials

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