Jianke Yang

General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation

General high-order rogue waves in the nonlinear Schrödinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N-th order rogue waves contain N−1 free irreducible complex parameters. In addition, the specific rogue waves obtained by Akhmediev et al. (Akhmediev et al. 2009 Phys. Rev. E 80, 026601 (doi:10.1103/PhysRevE.80.026601)) correspond to special choices of these free parameters, and they have the highest peak amplitudes among all rogue waves of the same order. If other values of these free parameters are taken, however, these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns.

Stability analysis for solitons in PT-symmetric optical lattices

Stability of solitons in parity-time (PT)-symmetric periodic potentials (optical lattices) is analyzed in both one- and two-dimensional systems. First we show analytically that when the strength of the gain-loss component in the PT lattice rises above a certain threshold (phase transition point), an infinite number of linear Bloch bands turn complex simultaneously. Second, we show that while stable families of solitons can exist in PT lattices, increasing the gain-loss component has an overall destabilizing effect on soliton propagation. Specifically, when the gain-loss component increases, the parameter range of stable solitons shrinks as new regions of instability appear. Third, we investigate the nonlinear evolution of unstable PT solitons under perturbations, and show that the energy of perturbed solitons can grow unbounded even though the PT lattice is below the phase transition point.

Rogue waves in the Davey-Stewartson I equation

General rogue waves in the Davey-Stewartson-I equation are derived by the bilinear method. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background with a line profile and then disappear into the constant background again. It is also shown that multirogue waves describe the interaction of several fundamental rogue waves. These multirogue waves also arise from the constant background and then decay back to it, but in the intermediate times, interesting curvy wave patterns appear. However, higher-order rogue waves exhibit different dynamics. Specifically, only part of the wave structure in the higher-order rogue waves rises from the constant background and then retreats back to it, and this transient wave possesses patterns such as parabolas. But the other part of the wave structure comes from the far distance as a localized lump, which decelerates to the near field and interacts with the transient rogue wave, and is then reflected back and accelerates to the large distance again.

Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared-operator method, the modified squared-operator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrodinger equations with periodic potentials, and isolated solitons in GinzburgLandau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.

Conditions for stationary pulse propagation in the strong dispersion management regime

Abstract Using the variational method, we obtain analytical conditions for stationary propagation of a Gaussian pulse in a fibre with strong dispersion management. We consider both the lossless fibre and the one with losses and periodic amplification. The analytical predictions have been checked against direct numerical simulations, and a good agreement between the two has been demonstrated. In particular, we find that in a certain region of parameters, the average dispersion necessary to support the stationary propagation is negative (normal). We also show that under a certain assumption, the variance of the Gordon-Haus timing jitter for the pulse in a strongly dispersion-managed system approximately equals that for the conventional soliton, reduced by an energy enhancement factor. Using our analytical conditions, we obtain an estimate for this factor. In particular, we show that in the presence of losses and periodic amplification, this jitter suppression factor can be made to be as large as that for the lossless case, by properly choosing the segment lengths in the dispersion map.

Dynamics of rogue waves in the Davey?Stewartson II equation

General rogue waves in the Davey–Stewartson (DS)II equation are derived by the bilinear method, and the solutions are given through determinants. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background in a line profile and then retreat back to the constant background again. It is also shown that multi-rogue waves describe the interaction between several fundamental rogue waves, and higher order rogue waves exhibit different dynamics (such as rising from the constant background but not retreating back to it). Under certain parameter conditions, these rogue waves can blow up to infinity in finite time at isolated spatial points, i.e. exploding rogue waves exist in the DSII equation.

Defect modes in one-dimensional photonic lattices.

Linear defect modes in one-dimensional photonic lattices are studied theoretically. For negative (repulsive) defects, various localized defect modes are found. The strongest confinement of the defect modes appears when the lattice intensity at the defect site is nonzero rather than zero. When launched at small angles into such a defect site of the lattice, a Gaussian beam can be trapped and undergo snake oscillations under appropriate conditions.

Classification of the solitary waves in coupled nonlinear Schro¨dinger equations

In this paper, the solitary waves in coupled nonlinear Schrodinger equations are classified into infinite families. For each of the first three families, the parameter region is specified and the parameter dependence of its solitary waves described and explained. We found that the parameter regions of these solution families are novel and irregular, and the parameter dependence of the solitary waves is sensitive. The stability of these families of solitary waves is also determined. We showed that only the family of symmetric and single-humped solitary waves is stable.

A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity

The Petviashvilis iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: -Mu+up=0, where M is a positive definite self-adjoint operator and p=const. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.

Dipole solitons in optically induced two-dimensional photonic lattices

Dipole solitons in a two-dimensional optically induced photonic lattice are theoretically predicted and experimentally demonstrated for the first time to our knowledge. It is shown that such dipole solitons are stable and robust under appropriate conditions. Our experimental results are in good agreement with theoretical predictions.