Modified nonlinearly dispersive mK(m,n,k) equations: I. New compacton solutions and solitary pattern solutions

In this paper exact solutions of the modified nonlinearly dispersive KdV equations (simply called mK(m,n,k) equations), u^m−1u_t+a(uⁿ)_x+b(u^k)_xxx=0, are investigated by using some direct ansatze. As a result, abundant new compacton solutions: solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave solutions and periodic wave solutions are obtained.     

The new extended Jacobian elliptic function expansion algorithm and its applications in nonlinear mathematical physics equations

More recently we have presented the extended Jacobian elliptic function expansion method and its algorithm to seek more types of doubly periodic solutions. Based on the idea of the method, by studying more relations among all twelve kinds of Jacobian elliptic functions. we further extend the method to be a more general method, which is still called the extended Jacobian elliptic function expansion method for convenience. The new method is more powerful to construct more new exact doubly periodic solutions of nonlinear equations. We choose the (2+1)-dimensional dispersive long-wave system to illustrate our algorithm. As a result, twenty-four families of new doubly periodic solutions are obtained. When the modulus m→1 or 0, these doubly periodic solutions degenerate as soliton solutions and trigonometric function solutions. This algorithm can be also applied to other nonlinear equations.     

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.     

Numerical stability of symmetric solitary-wave-like waves of a two-layer fluid—Forced modified KdV equation

Forced internal waves at the interface of a two-layer incompressible fluid in a two-dimensional domain with rigid horizontal boundaries are studied. The lower boundary is assumed to have a small obstruction. We derive a time-dependent forced modified KdV equation when the KdV theory fails and study the stabilities of four types of symmetric time-independent solitary-wave-like solutions numerically.     

Characterising the Martin-Löf random sequences using computably enumerable sets of measure one

A sequence ω is Martin-Löf random if and only if it appears early in every Lebesgue measure one set of computably enumerable intervals.     

Direct Integration of the Newton Potential over Cubes

In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain. In this paper we describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way. Received April 18, 2001; revised September 17, 2001 Published online April 25, 2002     

Solution of Polynomial Systems Derived from Differential Equations

Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions of the nonlinearity. However, in general, one cannot forecast how many solutions a boundary value problem may possess or even determine the existence of a solution. In recent years numerical continuation methods have been developed which permit the numerical approximation of all complex solutions of systems of polynomial equations. In this paper, numerical continuation methods are adapted to numerically calculate the solutions of finite difference discretizations of nonlinear two-point boundary value problems. The approach taken here is to perform a homotopy deformation to successively refine discretizations. In this way additional new solutions on finer meshes are obtained from solutions on coarser meshes. The complicating issue which the complex polynomial system setting introduces is that the number of solutions grows with the number of mesh points of the discretization. To counter this, the use of filters to limit the number of paths to be followed at each stage is considered.     

Liouvillian solutions of third order differential equations

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for four finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations.     

Extension of ENO and WENO schemes to one-dimensional sediment transport equations

Essentially nonoscillatory and weighted essentially nonoscillatory schemes are high order resolution schemes constructed for the hyperbolic conservation laws. In this paper we extend these schemes to the one-dimensional bed-load sediment transport equations. The difficulties that arise in the numerical modelling come from the fact that a nonconservative product is present in the system. Our specific numerical approximations for the nonconservative product are based on two ideas. First is to include the influence of that term in the system upwinding and the second is to define the numerical approximation in such a way that the obtained scheme solves the system for the quiescent flow case exactly. As a consequence, the resulting schemes give excellent results, as it can be seen from the numerical tests we present. On the opposite, the numerical results obtained by applying the pointwise evaluation of nonconservative product on the same tests present unacceptably large numerical errors.     

Minimal rational parametrizations of canal surfaces

It is well known that canal surfaces defined by a rational spine curve and a rational radius function are rational. The aim of the present paper is to construct a rational parameterization of low degree. The author uses the generalized stereographic projection in order to transform the problem to a parameterization problem for ruled surfaces. Two problems are discussed: parameterization with boundary conditions (design of canal surfaces with two curves on it, as is the case for rolling ball blends) and parameterization without boundary conditions.     

A New Sparse Gaussian Elimination Algorithm and the Niederreiter Linear System for Trinomials over F 2

An important factorization algorithm for polynomials over finite fields was developed by Niederreiter. The factorization problem is reduced to solving a linear system over the finite field in question, and the solutions are used to produce the complete factorization of the polynomial into irreducibles. One charactersistic feature of the linear system arising in the Niederreiter algorithm is the fact that, if the polynomial to be factorized is sparse, then so is the Niederreiter matrix associated with it. In this paper, we investigate the special case of factoring trinomials over the binary field. We develop a new algorithm for solving the linear system using sparse Gaussian elmination with the Markowitz ordering strategy. Implementing the new algorithm to solve the Niederreiter linear system for trinomials over F₂ suggests that, the system is not only initially sparse, but also preserves its sparsity throughout the Gaussian elimination phase. When used with other methods for extracting the irreducible factors using a basis for the solution set, the resulting algorithm provides a more memory efficient and sometimes faster sequential alternative for achieving high degree trinomial factorizations over F₂.     

Reaction diffusion equations and the chronification of liver infections

A predator-prey system is used to model the time-dependent virus and lymphocyte population during a liver infection. We show mathematically that the resulting reaction-diffusion equation has non-trivial stationary solutions whenever the underlying domain is sufficiently large or fissured. The non-trivial stationary solutions are interpreted as chronic liver infections. Thus qualitative differences between acute and chronic hepatitis infections become dispensable. Finally, numerical simulations for the chronification are presented.     

Geometric modeling of spatial constraints: objectives, methods and solid-modeling requirements

Robust Product Lifecycle Management (PLM) technology requires availability of informationally- complete models for all parts of a design-project including spatial constraints. This is the subject of the present investigation, leading to a new model for spatial constraints, the ``virtual solid', which generalizes a similar concept used by Sapidis and Theodosiou to model ``required free-spaces' in plants [14]. The present research focuses on the solid-modeling aspects of the virtual-solid methodology, and derives new solid-modeling problems (related to object definition and to object processing), whose robust treatment is a prerequisite for developing efficient models for complex spatial constraints.     

Implementations of a New Theorem for Computing Bounds for Positive Roots of Polynomials

Finding an upper bound for the positive roots of univariate polynomials is an important step of the continued fractions real root isolation algorithm. The revived interest in this algorithm has highlighted the need for better estimations of upper bounds of positive roots. In this paper we present a new theorem, based on a generalization of a theorem by D. Stefanescu, and describe several implementations of it – including Cauchy's and Kioustelidis' rules as well as two new rules recently developed by us. From the empirical results presented here we see that applying various implementations of our theorem – and taking the minimum of the computed values – greatly improves the estimation of the upper bound and hopefully that will affect the performance of the continued fractions real root isolation method.     

一般Hirota-Satsuma方程的多孤子解及孤子间的相互作用

用改进的齐次平衡法,首先把不可积的一般Hirota-Satsuma方程简化成可积模型—KdV方程,然后通过求解KdV方程得到了一般Hirota-Satsuma方程的多孤子解.利用得到的多孤子解分析了奇异孤子之间、钟型孤子与奇异孤子之间的相互作用,结果发现了相互作用的一些重要性质.     

Boolean sets and most general solutions of Boolean equations

The aim of this paper is twofold. First we determine the most general form of the subsumptive general solution of a Boolean equation (Theorem 1 and Theorem 2). Then we discuss several characterizations of Boolean sets, meaning sets of zeros of Boolean functions, and prove that every Boolean transformation X=Φ(T) is the parametric general solution of a certain Boolean equation.     

Generalized perturbed complex Toda chain for Manakov system and exact solutions of Bose–Einstein mixtures

We analyze, both analytically and numerically, the dynamical behavior of the N-soliton train in the Manakov system with perturbations due to an external potential. The perturbed complex Toda chain model has been employed to describe adiabatic interactions within the N Manakov-soliton train. Simulations performed demonstrate that external potentials can play a stabilizing role to provide bound state regime of the train propagation. We also present new stationary and travelling wave solutions to equations describing Bose–Fermi mixtures in an elliptic external potential. Precise conditions for the existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.     

Function-algebraic characterizations of log and polylog parallel time

The main results of this paper are recursion-theoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fan-in circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, anda priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the tiered recursion techniques of Leivant and Bellantoni & Cook.     

New soliton-like solutions and compacton-like periodic wave solutions of parametric type to a nonlinear dispersive equation

In this paper, by using the integral bifurcation method, we study a nonlinear dispersive equation. Some new soliton-like solutions and some compacton-like periodic wave solutions are obtained. Their dynamic characters are investigated and the profiles are given by the mathematical software Maple. From the graphs of some soliton-like solutions, we find that their profiles are transformable.