Obliquely propagating skew KP lumps

Obliquely propagating skew lumps are studied within the framework of the Kadomtsev–Petviashvili equation with a positive dispersion (the KP1 equation). Specific features of such lumps are analysed in detail. It is shown that skew multi-lump solutions can be also constructed within the framework of such an equation. As an example, the bi-lump solution is presented in the explicit form, analysed and illustrated graphically. The relevance of skew lumps to the real physical systems is discussed.     

Conservation laws,bilinear forms and solitons for a fifth-order nonlinear Schrödinger equation for the attosecond pulses in an optical fiber

Under investigation in this paper is a fifth-order nonlinear Schrödinger equation, which describes the propagation of attosecond pulses in an optical fiber. Based on the Lax pair, infinitely-many conservation laws are derived. With the aid of auxiliary functions, bilinear forms, one-, two- and three-soliton solutions in analytic forms are generated via the Hirota method and symbolic computation. Soliton velocity varies linearly with the coefficients of the high-order terms. Head-on interaction between the bidirectional two solitons and overtaking interaction between the unidirectional two solitons as well as the bound state are depicted. For the interactions among the three solitons, two head-on and one overtaking interactions, three overtaking interactions, an interaction between a bound state and a single soliton and the bound state are displayed. Graphical analysis shows that the interactions between the two solitons are elastic, and interactions among the three solitons are pairwise elastic. Stability analysis yields the modulation instability condition for the soliton solutions.     

关于修正的一维分子晶体模型的孤子激发

从立方型相互作用的一维分子晶体模型出发,在加入本征值平方项的近似下,运用能量极小原理和连续化近似,得到了考虑本征值平方项和立方型相互作用后一维分子晶体模型的孤子激发的修正解.在忽略了立方型相互作用和本征值平方项的近似下,孤子激发的修正解回复于通常的极化子解.并计算了孤子激发的峰宽、峰值和电子自陷势阱,分析了本征值平方项和立方型相互作用对孤子激发的峰值、峰宽和电子自陷势阱的影响.     

Multi-kink solutions and soliton fission and fusion of Sharma-Tasso-Olver equation

From the Levi spectral problem, we obtain two basic Darboux transformations of the Sharma-Tasso-Olver equation. Then from the trivial seed solution, we obtain multi-kink solutions and soliton fission and fusion solutions of this equation.     

One-dimensional polariton solitons and soliton waveguiding in microcavities

We extend our recent results [O.A. Egorov et al. Phys. Rev. Lett. 102, 153904 (2009)] on half-light–half-matter polariton solitons in planar semiconductor microcavities operating in the strong coupling regime. We initiate discussion on the structure of the solitons in the momentum space and its link to the instability of the upper branch of the polariton bistability loop. Numerical results showing the soliton excitation by a seed pulse are presented.     

Bright and dark solitons in a quasi-1D Bose-Einstein condensates modelled by 1D Gross-Pitaevskii equation with time-dependent parameters

We investigate the exact bright and dark solitary wave solutions of an effective 1D Bose-Einstein condensate (BEC) by assuming that the interaction energy is much less than the kinetic energy in the transverse direction. In particular, following the earlier works in the literature Pérez-García et al. (2004) [50], Serkin et al. (2007) [51], Gurses (2007) [52] and Kundu (2009) [53], we point out that the effective 1D equation resulting from the Gross-Pitaevskii (GP) equation can be transformed into the standard soliton (bright/dark) possessing, completely integrable 1D nonlinear Schrödinger (NLS) equation by effecting a change of variables of the coordinates and the wave function. We consider both confining and expulsive harmonic trap potentials separately and treat the atomic scattering length, gain/loss term and trap frequency as the experimental control parameters by modulating them as a function of time. In the case when the trap frequency is kept constant, we show the existence of different kinds of soliton solutions, such as the periodic oscillating solitons, collapse and revival of condensate, snake-like solitons, stable solitons, soliton growth and decay and formation of two-soliton bound state, as the atomic scattering length and gain/loss term are varied. However, when the trap frequency is also modulated, we show the phenomena of collapse and revival of two-soliton like bound state formation of the condensate for double modulated periodic potential and bright and dark solitons for step-wise modulated potentials.     

The improved Fan sub-equation method and its application to the SK equation

In this paper, many types of exact solutions of a first-order nonlinear ordinary differential equation called Fan sub-equation, is further investigated by using bifurcation method of dynamical systems. As a result, more types of exact solutions to Sawada-Kotera (SK) equation are obtained, which include more general soliton solutions, kink solutions, triangular function solutions, Jacobian elliptic function solutions with double periods and so on.     

Existence of gap solitons in periodic discrete nonlinear Schrödinger equations

In this paper, we discuss how to use the critical point theory to study the existence of gap solitons for periodic discrete nonlinear Schrödinger equations. An open problem proposed by Professor Alexander Pankov is solved.     

The homogeneous balance method and its application to the Benjamin-Bona-Mahoney (BBM) equation

We make use of the homogeneous balance method and symbolic computation to construct new exact traveling wave solutions for the Benjamin-Bona-Mahoney (BBM) equation. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

New exact solutions to some variable coefficients problems

With the aid of computer symbolic computation system such as Maple, an extended tanh method is applied to determine the exact solutions for some nonlinear problems with variable coefficients. Several new soliton solutions and periodic solutions can be obtained if we taking paraments properly in these solutions. The employed methods are straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.