GENERAL SOLITON SOLUTIONS OF THE COUPLED SCALAR FIELDS

Let σ and ρ be two field variables of the n-dimensional coupled scalar fields. Taking the function transformations σ=σ(ξ) and ρ=ρ(ξ) leads to an equation F(σ,ρ,dσ/dρ,d²σ/d²ρ) =0. Inserting any solution of this equation into the field equations yields a pair of general soliton solutions σ=σ(ξ) and ρ=ρ[σ(ξ)]. Some interesting specific soliton solutions are given. The stability and other properties of these solitons are discussed.     

Fusion,fission, and annihilation of complex waves for the （2＋l）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system

With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Fusion,fission, and annihilation of complex waves for the(2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff system

With the help of the symbolic computation system, Maple and Riccati equation( ξ= a0+ a1ξ+ a22ξ), expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx + my + nt + Γ(x, y,t) for the(2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff system(GCBS) are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Compacton-Like Solutions in a Camassa-Holm Type Equation

This paper shows that a Camassa-Hohn type equation can be reduced to Hamiltonian system by transformation of variables.The hamiltonian system is studied by making use of the dynamical systems theory and the qualitative behavior of degenerate singular points is presented.In particular,new type of compacton-like solutions is obtained by setting the partial differential equation under boundary condition limξ→±∞Ψ(ξ) = A.     

N-fold Darboux Transformation for Integrable Couplings of AKNS Equations

For the integrable couplings of Ablowitz-Kaup-Newell-Segur(ICAKNS) equations, N-fold Darboux transformation(DT) TN, which is a 4 × 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the(4N + 1)-order determinant and 4N-order determinant of eigenfunctions. By making use of these formulae,the determinant expressions of N-transformed new solutions p~([N ]), q~([N ]), r~([N ])and s~([N ])are generated by this N-fold DT.Furthermore, when the reduced conditions q =-p*and s =-r*are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schr?dinger(ICNLS) equations.Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example.     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f（ξ）, is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Novel traveling wave solutions and stability analysis of perturbed Kaup-Newell Schrodinger dynamical model and its applications

We study the traveling wave and other solutions of the perturbed Kaup-Newell Schrodinger dynamical equation that signifies long waves parallel to the magnetic field.The wave solutions such as bright-dark(solitons),solitary waves,periodic and other wave solutions of the perturbed Kaup-Newell Schrodinger equation in mathematical physics are achieved by utilizing two mathematical techniques,namely,the extended F-expansion technique and the proposed exp(-φ(ξ))-expansion technique.This dynamical model describes propagation of pluses in optical fibers and can be observed as a special case of the generalized higher order nonlinear Schrodinger equation.In engineering and applied physics,these wave results have key applications.Graphically,the structures of some solutions are presented by giving specific values to parameters.By using modulation instability analysis,the stability of the model is tested,which shows that the model is stable and the solutions are exact.These techniques can be fruitfully employed to further sculpt models that arise in mathematical physics.     

Second-order solutions for random interfacial waves in N-layer density-stratified fluid with steady uniform currents

In the present paper, the random interfacial waves in N-layer density-stratified fluids moving at different steady uniform speeds are researched by using an expansion technique, and the second-order asymptotic solutions of the random displacements of the density interfaces and the associated velocity potentials in N-layer fluid are presented based on the small amplitude wave theory. The obtained results indicate that the wave-wave second-order nonlinear interactions of the wave components and the second-order nonlinear interactions between the waves and currents are described. As expected, the solutions include those derived by Chen （2006） as a special case where the steady uniform currents of the N-layer fluids are taken as zero, and the solutions also reduce to those obtained by Song （2005） for second-order solutions for random interfacial waves with steady uniform currents if N = 2.     

Invariant Sets and Exact Solutions to Higher-Dimensional Wave Equations

The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

Abstract In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f （ξ）, is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

Exact Solutions of the Two-Dimensional Cubic-Quintic Nonlinear Schrdinger Equation with Spatially Modulated Nonlinearities

Applying the similarity transformation,we construct the exact vortex solutions for topological charge S ≥ 1 and the approximate fundamental soliton solutions for S = 0 of the two-dimensional cubic-quintic nonlinear Schrdinger equation with spatially modulated nonlinearities and harmonic potential.The linear stability analysis and numerical simulation are used to exam the stability of these solutions.In different profiles of cubic-quintic nonlinearities,some stable solutions for S ≥ 0 and the lowest radial quantum number n = 1 are found.However,the solutions for n ≥ 2 are all unstable.     

Soliton, Positon and Negaton Solutions of Extended KdV Equation

Darboux transformation （DT） provides us with a comprehensive approach to construct the exact and explicit solutions to the negative extended KdV （eKdV） equation, by which some new solutions such as singular soliton, negaton, and positon solutions are computed for the eKdV equation. We rediscover the soliton solution with finiteamplitude in [A.V. Slyunyaev and E.N. Pelinovskii, J. Exp. Theor. Phys. 89 （1999） 173] and discuss the difference between this soliton and the singular soliton. We clarify the relationship between the exact solutions of the eKdV equation and the spectral parameter. Moreover, the interactions of singular two solitons, positon and negaton, positon and soliton, and two positons are studied in detail.     

Some New Solutions of Jacobi Elliptic Functions to mBBM Equation

The modified mapping method is further improved by the expanded expression of u（ξ） that contains the terms of the first-order derivative of function f（ξ）. Some new exact solutions to the mBBM equation are determined by means of the method. We can obtain many new solutions in terms of the Jacobi elliptic functions of the equation.     

New Types of Exact Solutions for (N 1)-Dimensional φ4 Model

New types of exact solutions of the (N 1)-dimensional φ4-model are studied in detail. Some types of interaction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interaction solutions are found.     

Collisional Efect on Weibel Instability with Semi-Relativistic Maxwellian Distribution Function

In this paper,the Coulomb collisional efect of electron-ion on the growth rate of Weibel instability is investigated based on the semi-relativistic Maxwellian distribution function in dense and unmagnetized plasma.An analytical expression was derived for the dispersion relation of Weibel instability for two limit cases|ξ=ω′/k|| T|||≥1 and |ξ|≤1.In limit|ξ|≥1 the dispersion relation only includes a real part and in limit|ξ|≤1 the imaginary part of the frequency of waves’instability plays a role in the dispersion relation.In limit≤|ξ|1,the two quantitiesμandη,that are due to the relativistic and collisional efects,will appear in the growth rate of Weibel instability.The growth rate of Weible istability will be increased through decreasing the Coulomb collisional frequency and also increasing the temperature anisotropic parameter in strong relativistic limit.     

Rational Solutions for the Fokas System

Fokas system is the simplest(2+1)-dimensional extension of the nonlinear Schrodinger equation(Eq.(2),Inverse Problems 10(1994) L19-L22).By using the bilinear transformation method,general rational solutions for the Fokas system are given explicitly in terms of two order-N determinants T_n(n = 0,1) whose elements m_(i,j)~(n)(n = 0,1;1≤i,j≤N)are involved with order-n_i and order-n_j derivatives.When N = 1,three kinds of rational solution,i.e.,fundamental lump and fundamental rogue wave(RW) with n_1 = 1,and higher-order rational solution with n_1 2,are illustrated by explicit formulas from T_n(n = 0,1) and pictures.The fundamental RW is a line RW possessing a line profile on(x,y)-plane,which arises from a constant background with at t 0 and then disappears into the constant background gradually at t 0.The fundamental lump is a traveling wave,which can preserve its profile during the propagation on(x,y)-plane.When N ≥2 and n_1 =n_2=...=n_n = 1,several specific multi-rational solutions are given graphically.     

Analysis of Bd→ψ KS CP Asymmetry in a Flavour Changing Z'' Model

According to the recent experimental data, the time-dependent CP asymmetry SφKs for Ba → CKs decay might still be inconsistent with the standard model （SM） expectations. We try to understand the anomaly with a Z＇ model associated with flavour changing neutral currents （FCNCs） at the tree level. We find that （i） if neglecting the contributions from the right-handed flavour changing coupling Bsb^R and setting ξ^LL = ξ^LR =ξ and ξ^LL = ξ, ξ^LR = 0, we may obtain 0.01 〈ξ 〈 0.014, 0.0175 〈ξ〈 0.0205, 0.021 〈 ξ 〈 0.0255 and 0.004 〈 ξ 〈 0.008, 0.018 〈ξ 〈 0.028, 0.042 〈ξ〈 0.048 for the Bd → φKs decay, respectively; （ii） if considering the contributions coming from Bsb^R and setting ξ^LL = ξ^RL= ξ^LR = ξ,ξ^LR =ξ^RR=0, we can obtain 0.005 〈 ξ 〈 0.075, 0.0085 〈 ξ 〈 0.0105, 0.011 〈 ξ 〈 0.013 and 0.0125 〈 ξ 〈 0.0177, 0.0224 〈 ξ 〈 0.025. These results are consistent with the constraints and the assumptions in some literature.     

New Types of Exact Solutions for （N＋1）-Dimensional φ^4 Model

New types of exact solutions of the （N ＋ 1）-dimensional φ^4-model are studied in detail. Some types of interaction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interaction solutions are found.     

Symbolic Computation and Construction of Soliton－Like Solutions to the（2＋l）－Dimensional Breaking Soliton Equation

Based on the computerized symbolic system Mapte, a new generalized expansion method of Riccati equation for constructing non-travelling wave and coefficient functions‘ soliton-like solutions is presented by a new general ansatz. Making use of the method, we consider the (2 1)-dimensional breaking soliton equation, ut buxxy 4buvx 4buxv = O,uv=vx, and obtain rich new families of the exact solutions of the breaking sofiton equation, including then on-traveilin~ wave and constant function sofiton-like solutions, singular soliton-like solutions, and triangular function solutions.     

Transportation Characteristics for a Class of Generalized N-Diffusion Equation with Convection

A suitable similarity transformation group is used to reduce the power law parabolic partial equation to a classof singular nonlinear boundary value problems. Analytical solutions and numerical solutions both are presented for the specific power law index N, conductivity and convective functions, and the associated transportation characteristics are discussed in detail.