Analytical Descriptions of Dark and Gray Solitons in Nonlocal Nonlinear Media

By using the standard symmetry reduction method,the gray/dark solitons and periodic waves(gray/dark soliton lattice) are analytically studied for the nonlinear optical media with periodic nonlocal response.It is found that there are two critical points for the quantity β=ω_m~2/ω_0~2,the multiplication of the square of the wave number(1/ω_0)and the strength(w_m~2) of the nonlocality both for the soliton and periodic solutions.The soliton solution exists only for β≤ 1/4 and the soliton is a double well gray soliton for β 1/8 while it is a single well gray soliton for β≤ 1/8.The soliton is dark only for β=1/4,otherwise it is a gray soliton.Similar critical points exist for the gray/dark soliton lattice solutions.     

Variable Separation Solution for （1＋1）-Dimensional Nonlinear Models Related to Schroedinger Equation

A variable separation approach is proposed and successfully extended to the (1 1)-dimensional physics models. The new exact solution of (1 1)-dimensional nonlinear models related to Schr6dinger equation by the entrance of three arbitrary functions is obtained. Some special types of soliton wave solutions such as multi-soliton wave solution,non-stable soliton solution, oscillating soliton solution, and periodic soliton solutions are discussed by selecting the arbitrary functions appropriately.     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion(CTE) method is applied to the(2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution,and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlev′e truncated expansion method.And we investigate interactive properties of solitons and periodic waves.     

Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.     

Pseudo analytical solution to time periodic stiffness systems

An analytical form of state transition matrix for a system of equations with time periodic stiffness is derived in order to solve the free response and also allow for the determination of system stability and bifurcation. A pseudoclosed form complete solution for parametrically excited systems subjected to inhomogeneous generalized forcing is developed,based on the Fourier expansion of periodic matrices and the substitution of matrix exponential terms via Lagrange-Sylvester theorem. A Mathieu type of equation with large amplitude is presented to demonstrate the method of formulating state transition matrix and Floquet multipliers. A two-degree-of-freedom system with irregular time periodic stiffness characterized by spiral bevel gear mesh vibration is presented to find forced response in stability and instability. The obtained results are presented and discussed.     

Bifurcation analysis of the logistic map via two periodic impulsive forces

The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincare′ map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.     

Current–phase relations of a ring-trapped Bose–Einstein condensate with a weak link

The current–phase relations of a ring-trapped Bose–Einstein condensate interrupted by a rotating rectangular barrier are extensively investigated with an analytical solution. A current–phase diagram, single and multi-valued relation, is presented with a rescaled barrier height and width. Our results show that the finite size makes the current–phase relation deviate a little bit from the cosine form for the soliton solution in the limit of a vanishing barrier, and the periodic boundary condition selects only the plane wave solution in the case of high barrier. The reason for multi-valued current–phase relation is given by investigating the behavior of soliton solution.     

Bcklund Transformations and Solutions of a Generalized Kadomtsev-Petviashvili Equation

In this paper,the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials.The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function,respectively.And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution.In the end,the bilinear Bcklund transformations are derived.     

Soliton–cnoidal interactional wave solutions for the reduced Maxwell–Bloch equations

In this paper, we study soliton–cnoidal wave solutions for the reduced Maxwell–Bloch equations. The truncated Painlev′e analysis is utilized to generate a consistent Riccati expansion, which leads to solving the reduced Maxwell–Bloch equations with solitary wave, cnoidal periodic wave, and soliton–cnoidal interactional wave solutions in an explicit form.Particularly, the soliton–cnoidal interactional wave solution is obtained for the first time for the reduced Maxwell–Bloch equations. Finally, we present some figures to show properties of the explicit soliton–cnoidal interactional wave solutions as well as some new dynamical phenomena.     

Exact Periodic Wave Solution of Extended(2+1)-Dimensional Shallow Water Wave Equation with Generalized D_-operators

With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.