(2+1)维Korteweg-de Vries方程的传播孤子及混沌行为

利用一个投射方程和变量分离法,得到了(2+1)维Korteweg-de Vries(KdV)方程的新显式精确解.根据得到的孤立波解,构造出KdV方程的传播孤子结构.利用一个新的混沌系统研究了孤子的混沌行为。     

Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos

In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitary-wave-like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. Firstly, it allows us to identify the stable solution of the stochastic governing equation. Secondly, it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves.     

A Boussinesq system for two-way propagation of interfacial waves

The theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering, and as a source of interesting mathematical model equations that exhibit nonlinearity and dispersion. A Boussinesq system for two-way propagation of interfacial waves in a rigid lid configuration is derived. In most cases, the nonlinearity is quadratic. However, when the square of the depth ratio is close to the density ratio, the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered. The propagation as well as the collision of solitary waves and/or fronts is studied numerically.     

Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg-de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations

We apply the Simple Equations Method (SEsM) for obtaining exact travelling-wave solutions of the extended fifth-order Korteweg-de Vries (KdV) equation. We present the solution of this equation as a composite function of two functions of two independent variables. The two composing functions are constructed as finite series of the solutions of two simple equations. For our convenience, we express these solutions by special functions V, which are solutions of appropriate ordinary differential equations, containing polynomial non-linearity. Various specific cases of the use of the special functions V are presented depending on the highest degrees of the polynomials of the used simple equations. We choose the simple equations used for this study to be ordinary differential equations of first order. Based on this choice, we obtain various travelling-wave solutions of the studied equation based on the solutions of appropriate ordinary differential equations, such as the Bernoulli equation, the Abel equation of first kind, the Riccati equation, the extended tanh-function equation and the linear equation.     

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

In this paper, an infinite sequence of conservation laws for a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids are constructed based on the Bäcklund transformation. Hirota bilinear form and symbolic computation are applied to obtain three kinds of solutions. Variablecoefficients can affect the conserved density, associated flux, andappearance of the characteristic lines. Effects of the wave number on the soliton structures are also discussed and types of soliton structures, e.g., the double-periodic soliton, parallel soliton and soliton complexes, are presented.     

Bifurcations in fluctuating systems: The center-manifold approach

Bifurcations in fluctuating dynamical systems are studied using the ideas of center-manifold reduction. The method provides not only a systematic procedure for the reduction of the system to a small number of variables-but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points. The joint probability density factorizes into a stationary Gaussian densityp(v/u) in the fast variablesv, and a time-dependent densityP(u, f) in the slow variablesu describing the dynamics on the center manifoldv=v₀ (u). P(u, t) obeys a reduced Fokker-Planck equation that can be written in a normal form by means of local nonlinear transformations. Both additive and multiplicative white noise are considered, as is colored noise. The results extend and formalize Haken's concept of adiabatic elimination of fast variables.     

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits.     

A lattice of electromagnetic solitons in a superlattice formed by a system of quantum dots

Considering the Maxwell equations for the electromagnetic-field propagation in a solid with a three-dimensional superlattice of quantum dots linked by strong tunneling along one axis, we obtained a phenomenological equation in the form of the classical 2+1-dimensional sine-Gordon equation. Electrons were considered classically in the formalism of the Boltzmann kinetic equation for the distribution function. Solutions were obtained as a soliton lattice for the vector potential of the electric field. These lattices emerge as a consequence of the coherent change of the classical distribution function and the electric field generated by tunneling electrons in a system of quantum wells.     

Breathers and solitons on two different backgrounds in a generalized coupled Hirota system with four wave mixing

We study breathers and solitons on different backgrounds in optical fiber system, which is governed by generalized coupled Hirota equations with four wave mixing effect. On plane wave background, a transformation between different types of solitons is discovered. Then, on periodic wave background, we find breather-like nonlinear localized waves of which formation mechanism are related to the energy conversion between two components. The energy conversion results from four wave mixing. Furthermore, we prove that this energy conversion is controlled by amplitude and period of backgrounds. Finally, solitons on periodic wave background are also exhibited. These results would enrich our knowledge of nonlinear localized waves' excitation in coupled system with four wave mixing effect.     

A semi-classical approach to two-frequency solitons in a three-level cascade atomic system

A semi-classical theory of two intense optical fields interacting with a third-order non-linear medium composed of a three-level cascade atomic system is presented. It is predicted that non-linear atom-field interactions allow the formation of two-frequency bright, dark and grey spatial solitons. We demonstrate through numerical simulations and analytic stability analysis that the bright and grey solitons are stable.     

A kind of extended Korteweg——de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

This paper considers interfacial waves propagating along the interface between a two-dimensional two-fluid with a flat bottom and a rigid upper boundary. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. It just focuses on the weakly non-linear small amplitude waves by introducing two small independent parameters: the nonlinearity ratio $\varepsilon $, represented by the ratio of amplitude to depth, and the dispersion ratio $\mu $, represented by the square of the ratio of depth to wave length, which quantify the relative importance of nonlinearity and dispersion. It derives an extended KdV equation of the interfacial waves using the method adopted by Dullin {\it et al} in the study of the surface waves when considering the order up to $O(\mu ^2)$. As expected, the equation derived from the present work includes, as special cases, those obtained by Dullin {\it et al} for surface waves when the surface tension is neglected. The equation derived using an alternative method here is the same as the equation presented by Choi and Camassa. Also it solves the equation by borrowing the method presented by Marchant used for surface waves, and obtains its asymptotic solitary wave solutions when the weakly nonlinear and weakly dispersive terms are balanced in the extended KdV equation.     

Estimating parameters in stochastic systems: A variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.     

Diffusion in a symmetric bistable potential: A variational approach

An approximation procedure for the solution of stochastic nonlinear equations, which was derived from a variational principle in a previous paper, is applied to the problem of a particle that diffuses in a symmetric bistable potential starting from the point of unstable equilibrium. The second moment and variance for the particle's position are calculated as functions of the timet. Good agreement is found with results recently obtained by Baibuzet al. from an approximate evaluation of a path integral expression for the probability density.     

Electron correlations and quantum lattice vibrations in strongly coupled electron-phonon systems: A variational slave boson approach

We investigate the ground-state properties of the two-dimensional Hubbard model with an additional Holstein-type electron-phonon coupling on a square lattice. The effects of quantum lattice vibrations on the strongly correlated electronic system are treated by means of a variational squeezed-polaron wave function proposed by Zheng, where the possibility of static (frozen) phonon-staggered ordering is taken into account. Adapting the Kotliar-Ruckenstein slave boson approach to the effective electronic Hamiltonian, which is obtained in the vacuum state of the transformed phonon subsystem, our theory is evaluated within a two-sublattice saddle-point approximation at arbitrary band-filling over a wide range of electron-electron and electron-phonon interaction strengths. We determine the order parameters for long-range charge and/or spin ordered states from the self-consistency conditions for the auxilary boson fields, including an optimization procedure with respect to the variational displacement, polaron and squeezing parameters. In order to characterize the crossover from the adiabatic (=0) to the nonadiabatic (=) regime, the frequency dependencies of these quantities are studied in detail. In the predominant charge (spin) ordered phases the static Peierls dimerization (magnetic order) is strongly reduced with increasing . As the central result we present the slave boson ground-state phase diagram of the Holstein-Hubbard model for finite phonon frequencies.     

Bilinear forms through the binary Bell polynomials,N solitons and Bäcklund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-Bäcklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and Bäcklund transformations are different from those in the existing literature. All of our results are dependent on the water-wave dispersive power.