A correction of the 2D KdV equation of Djordjevic ＆ Redekopp in exponentially stratified fluid＊

To further study the fission laws of initial internal solitons on the continental shelf/slope, we rederive and correct the 2D KdV equation of Djordjevic & Redekopp for exponentially stratified fluid (or ocean) and with two-dimensional topography. Through a combination of theoretical study and numerical experiments, we show that solitons in the odd vertical modes can fission. However, because of the corrections, the fission conditions are different from those of Djordjevic & Redekopp. The even modes cannot fission unless the initial internal solitons propagate from shallow sea to deep sea. This conclusion is entirely opposite to that of Djordjevic & Redekopp.The project supported by the National Natural Science Foundation of China (40276008) and the Grant of Key Laboratory of Marine Science and Numerical Modeling, SOA (0201(2003))The English text was polished by Keren Wang.     

Non-autonomous bright matter wave solitons in spinor Bose–Einstein condensates

We investigate the dynamics of bright matter wave solitons in spin-1 Bose–Einstein condensates with time modulated nonlinearities. We obtain soliton solutions of an integrable autonomous three-coupled Gross–Pitaevskii (3-GP) equations using Hirota?s method involving a non-standard bilinearization. The similarity transformations are developed to construct the soliton solutions of non-autonomous 3-GP system. The non-autonomous solitons admit different density profiles. An interesting phenomenon of soliton compression is identified for kink-like nonlinearity coefficient with Hermite–Gaussian-like potential strength. Our study shows that these non-autonomous solitons undergo non-trivial collisions involving condensate switching.     

Peregrine solitons and algebraic soliton pairs in Kerr media considering space–time correction

Exact unified rational solutions describing a family of Peregrine solitons as well as related algebraic soliton pairs in either self-focusing or self-defocusing Kerr media are presented, with distinct defining regimes and an explicit relationship for those of opposite nonlinearity. The active role of the space–time correction effect that plays in these soliton species is highlighted, leading to unique dynamics such as the persistence of Peregrine solitons in a defocusing Kerr medium, the availability of giant peak amplitude for the bright–bright soliton pair, and the existence of so-called bright–dark soliton pair. The evolution dynamics of the algebraic two-soliton state toward a spliced single soliton is also discussed.     

Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation

We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton.     

Study of femtosecond soliton dynamics in photonic crystal fiber using the moment method

We investigate the dynamics of femtosecond solitons in photonic crystal fibers (PCFs) by including high-order dispersion terms until to sixth-order in the generalized nonlinear Schrödinger equation, in addition to the nonlinear effects of the self phase modulation, self steepening and Raman scattering. We calculate theoretically the pulse parameters using the moment method. In the case of the fundamental soliton, our computed equations are coupled and difficult to solve analytically. However, we use the finite difference method to calculate numerically pulse parameters using an initially hyperbolic secant pulse at 1550-nm with different peak powers along 10m-PCF. Our numerical results show that the nonlinear regimes allow obtaining pulse compressions and initial pulse amplitudes. Furthermore, we remark a pulse broadening, and weak shifts of the peak power positions and frequencies in the critical and dispersive regimes. The use of an initial chirp provides a better pulse compressions and especially for low input powers. Also, the initial positive chirp reduces the optimal compression position lengths, while the negative one increases them. Therefore, we conclude that our theoretical calculations and numerical simulation results show that the moment method associated with the finite differences method is effective for the study of femtosecond pulse dynamics in PCFs.     

Effect of line coding on jitter and initial laser phase on Q value in soliton transmission

In this paper we described how different line coding techniques vary with values of jitter in soliton transmission system. Then the idea of this soliton transmission system is applied for multiuser structure and effects of initial line phase spacing are examined.     

Bound states of solitons in a fiber laser mode locked with carbon nanotube saturable absorber

We report on the experimental observation of bound states of solitons in an erbium-doped fiber laser passively mode-locked by the carbon nanotube saturable absorber. Bound states of solitons with various pulse separations are obtained. While the tightly bound solitons always exhibit the same set of fixed discrete pulse separations with π or π/2 phase difference, the feature becomes less obvious for the loosely bound solitons. The result that various states of the bound solitons were obtained in the same fiber laser makes a systematic experimental study on them and a comparison of their properties possible. Our results once again show that the bound soliton emission is an intrinsic feature of the mode-locked soliton fiber lasers.     

Surface solitons in trilete lattices

Fundamental solitons pinned to the interface between three semi-infinite one-dimensional nonlinear dynamical chains, coupled at a single site, are investigated. The light propagation in the respective system with the self-attractive on-site cubic nonlinearity, which can be implemented as an array of nonlinear optical waveguides, is modeled by the system of three discrete nonlinear Schrödinger equations. The formation, stability and dynamics of symmetric and asymmetric fundamental solitons centered at the interface are investigated analytically by means of the variational approximation (VA) and in a numerical form. The VA predicts that two asymmetric and two antisymmetric branches exist in the entire parameter space, while four asymmetric modes and the symmetric one can be found below some critical value of the inter-lattice coupling parameter—actually, past the symmetry-breaking bifurcation. At this bifurcation point, the symmetric branch is destabilized and two new asymmetric soliton branches appear, one stable and the other unstable. In this area, the antisymmetric branch changes its character, getting stabilized against oscillatory perturbations. In direct simulations, unstable symmetric modes radiate a part of their power, staying trapped around the interface. Highly unstable asymmetric modes transform into localized breathers traveling from the interface region across the lattice without significant power loss.     

GLOBAL DYNAMICS OF DISSIPATIVE GENERALIZED KORTEWEG-DE VRIES EQUATIONS

§1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　　ＳｉｎｃｅｔｈｅｄｉｓｃｏｖｅｒｙｂｙＫｏｒｔｅｗｅｇａｎｄｄｅＶｒｉｅｓ(1895)ａｎｄｔｈｅｗｏｒｋｏｆＺａｂｕｓｋｙａｎｄＫｒｕｓｔａｌ(1965),ｔｈｅｒｅｈａｖｅｂｅｅｎｎｕｍｅｒｏｕｓｓｉｇｎｉｆｉｃａｎｔｃｏｎｔｒｉｂｕｔｉｏｎｓｔｏｔｈｅＫｏｒｔｅｗｅｇｄｅＶｒｉｅｓ(ＫｄＶ)ｅｑｕａｔｉｏｎｓａｎｄｔｈｅｓｏｌｉｔｏｎｓｏｌｕｔｉｏｎｓ,ｅｓｐｅｃｉａｌｌｙｔｈｅｍｅｔｈｏｄｏｆＩｎｖｅｒｓｅＳｃａｔｔｅｒｉｎｇＴｒａｎｓｆｏｒｍａｎｄｔｈｅｍｅｔｈｏｄｏｆ…