On the rigidity theorems for lagrangian translating solitons in pseudo-Euclidean space I

Let f be a smooth strictly convex solution of
defined on a domain Ω C R^n, where ai, bi and c are constants. Then the graph Mvf of △f is a spacelike translating soliton for mean curvature flow in pseudo-Euclidean space with the translating vector（al, a2, . ., an; bz, b2, , bn）. In this paper, we will use alCfine technique to show a Berustein Theorem： if the graph Mvf is complete, then f（x） must be a quadratic polynomial and Mvf is an ailine n-plane.     

Propagation of an electromagnetic soliton in an anisotropic biquadratic ferromagnetic medium

Information storage technology based on anisotropic ferromagnets with sufficiently high magneto-optical effects has received much attention in recent years.Magneto-optical recording combines the merits of magnetic and optical techniques.We investigate the magneto-optical effects on a biquadratic ferromagnet and show that the dynamics of the system are governed by a perturbed nonlinear Schro¨dinger equation.The evolutions of amplitude and velocity of the soliton are found to be time independent,thereby admitting the lossless propagation of the electromagnetic soliton in the medium,which may have potential applications in soliton based optical communication systems.We also exploit the role of perturbation,which has a significant impact on the propagation of an electromagnetic soliton.     

Superposition solitons in two-component Bose-Einstein condensates

We develop the Hirota bilinear method and obtain the exact one and two superposition soliton solutions for two-component Bose-Einstein condensates. The conversion of three kinds of solitons including the superposition solitons, bright-bright solitons, and dark-bright solitons is discussed. With the energy analysis, we find that the superposition soliton state is an excitation state for this system. Moreover, the collision of two superposition solitons is found to be elastic.     

Analytical and numerical solutions to the Davey–Stewartson equation with power-law nonlinearity

This paper studies the Davey–Stewartson equation. The traveling wave solution of this equation is obtained for the case of power-law nonlinearity. Subsequently, this equation is solved by the exponential function method. The mapping method is then used to retrieve more solutions to the equation. Finally, the equation is studied with the aid of the variational iteration method. The numerical simulations are also given to complete the analysis.     

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

In this paper, by using the bifurcation method of dynamical systems, we derive the traveling wave solutions of the nonlinear equation UU_τyy ? U_yU_τy + U²U_τ + 3U_y = 0. Based on the relationship of the solutions between the Novikov equation and the nonlinear equation, we present the parametric representations of the smooth and nonsmooth soliton solutions for the Novikov equation with cubic nonlinearity. These solutions contain peaked soliton, smooth soliton, W-shaped soliton and periodic solutions. Our work extends some previous results.     

Solitons in a generalized space- and time-variable coefficients nonlinear Schrödinger equation with higher-order terms

We introduce an approach that combines a similarity method with several transformations to find analytical solitary wave solutions for a generalized space- and time-variable coefficients of nonlinear Schrödinger equation with higher-order terms with consideration of varying dispersion, higher nonlinearities, gain/loss and external potential. One of these transformations is constructed in such a way that allows study of the width of localized solutions. Solitary-like wave solutions for front, bright and dark are given. The precise expressions of the soliton?s width, peak, and the trajectory of its mass center and the external potential which are symbol of dynamic behavior of these solutions, are investigated analytically. In addition, the dynamical behavior of moving, periodic, quasi-periodic of breathing, and resonant are discussed. Stability of the obtained solutions is analyzed both analytically and numerically.     

Dissipative solitons with energy and matter flows: Fundamental building blocks for the world of living organisms

We consider a combined model of dissipative solitons that are generated due to the balance between gain and loss of energy as well as to the balance between input and output of matter. The system is governed by the generic complex Ginzburg–Landau equation, which is coupled to a common reaction–diffusion (RD) system. Such a composite dynamical system may describe nerve pulses with a significant part of electromagnetic energy involved. We present examples of such composite dissipative solitons and analyse their internal balances between energy and matter generation and dissipation.     

Exact solution of the cubic-quintic Duffing oscillator

In this paper, we derive the exact solution of the cubic-quintic Duffing oscillator based on the use of Jacobi elliptic functions. We also showed that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind.     

Existence of bounds for dark solitons in a class of nonlinear Schrödinger equations: explicit upper and lower bounds

The dark soliton solutions corresponds to certain heteroclinic orbits for related autonomous planar differential equations. In this paper, we give a proof of the existence of upper and lower bounds of these heteroclinic orbits and find these bounds of explicit form. We put it into practice for a particular case of the derivative nonlinear Schrödinger equation. Copyright © 2012 John Wiley & Sons, Ltd.     

Scattering effect of atoms through a Bose-Einstein condensate in an optical Lattice with a single defect

We study the transport of atoms across a localized Bose-Einstein condensate in an onedimensional optical lattice with a single defect. Our analytical and numerical results show that the defect as well as the nonlinear parameter can control the transmission of the atoms beam and the position of total reflection caused by Fano resonance. These interesting features may be a very useful basis for devising tunable atom filters or a button.