New explicit nonperiodic solutions of the homogeneous wave equation

We exhibit a newansatz for the solution of the homogeneous three-dimensional time-dependent wave equation in spherical coordinates of the form Φ(r,t)=Y(θ, φ)(I(r)+G(g)), whereg ≡ct/r. FunctionG(g) has explicit solution in terms of three independent nonperiodic functionss _ℓ,t _ℓ,u _ℓ (s _ℓ andt _ℓ are related to the associated Legendre functions of the first and second kinds).G(g) is nonperiodic and may be cast as a superposition of incoming and outgoing waves. To obtainG(g), we solved a nonhomogeneous associated Legendre equation (this solution, to our knowledge, is also new).G(g) may prove useful in many microscopic and macroscopic problems, representable by homogeneous wave equations.     

The higher order Hamiltonian structures for the modified classical Yang-Baxter equation

We consider constructing the higher order Hamiltonian structures on the dual of the Lie algebra from the first Hamiltonian structure of the coadjoint orbit method. For this purpose we show that the structure of the Lie algebrag is inherited to the algebra of vector fields ong ^* through the solution of the Modified Classical Yang-Baxter equation (Classicalr matrix). We study the algebra that generates the compatible Poisson brackets.This work was supported by Grant Aid for Scientific Research, the Ministry of Education.     

Fifth order evolution equation for long wave dissipative solitons

Third and fifth order nonlinear wave equations which arise in the theory of water waves possess solitary and periodic traveling waves. Solitary waves also arise in systems with dissipation and instability where a balance between these effects allows the existence of dissipative solitons. Here we search for a model equation to describe long wave dissipative solitons including fifth order dispersion. The equation found includes quadratic and cubic nonlinearities. For periodic solutions in a small box we characterize the rate of growth, and show that they do not blow up in finite time. Analytic solutions are constructed for special parameter values.     

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation

The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.     

Finite difference methods for second order in space,first order in time hyperbolic systems and the linear shifted wave equation as a model problem in numerical relativity

Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems. Particular attention is paid to the case when first order derivatives that can be identified with advection terms are approximated with non-centered finite difference operators. We first derive general properties of these discrete operators, then we extend a known result on numerical stability for such systems to general order of accuracy. As an application we analyze the shifted wave equation, including the behavior of the numerical phase and group speeds at different orders of approximations. Special attention is paid to when the use of off-centered schemes improves the accuracy over the centered schemes.     

Higher order parabolic approximations of the reduced wave equation

Asymptotic solutions of order k⁻ⁿ are developed for the reduced wave equation. Here k is a dimensionless wave number and n is the arbitrary order of the approximation. These approximations are an extension of geometric acoustics theory and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations. These corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray. The theory is applied to the example of radiation from a piston, and it is demonstrated that the higher order approximations are more accurate for decreasing values of k.     

New types of solitary wave solutions for the higher order nonlinear Schrodinger equation

We present new types of solitary wave solutions for the higher order nonlinear Schrodinger (HNLS) equation describing propagation of femtosecond light pulses in an optical fiber under certain parametric conditions. Unlike the reported solitary wave solutions of the HNLS equation, the novel ones can describe bright and dark solitary wave properties in the same expressions and their amplitude may approach nonzero when the time variable approaches infinity. In addition, such solutions cannot exist in the nonlinear Schrodinger equation. Furthermore, we investigate the stability of these solitary waves under some initial pertubations by employing the numerical simulation methods.     

A non-relativistic wave equation of second order in time for scalar fields

In this paper a generalization of the traditional non-relativistic Schrödinger equation is considered. It is a wave equation of second order in time and fourth order in the space coordinates for scalar fields. The equation has certain features, which make it a closer analogue of the Klein-Gordon equation than the traditional Schrödinger equation. However, the equation maintains the non-relativistic relation between energy and momentum.The implications of this generalized wave equation and the quantized field theory based on it are studied. The theory can be shown to be charge symmetric and allows to introduce anti-particles and pair creation. We compare the Green functions for this theory with those of conventional non-relativistic quantum theory.The theory allows to formulate a transformation for charge conjugation. The PCT-theorem is valid for it. The usual spin-statistics connection holds.     

New solutions of a higher order wave equation of the KdV type

Abstract

In this paper we use the Painlevé analysis and study a special case of a water wave equation of the KdV type. More specifically, we use the Pickering algorithm [9] and obtain a new kind of solutions, which constitute of both algebraic and trigonometric (or hyperbolic) functions.     

Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary conditions on a line segment. This permits us to take advantage of Wronski's construction method for Green functions without knowledge of eigenvalues. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2 × 2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with the second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of this classical solution. In the presence of a zero mode, our method allows for a simple universal regularization of the functional determinants. For Dirichlet's boundary condition, our result is equivalent to Gelfand-Yaglom's. Explicit formulas are given for a harmonic oscillator with an arbitrary time-dependent frequency.     

Some explicit formulas of Lyapunov exponents for three-dimensional quadratic mappings

This paper shows that there exist six different cases where it is possible to find rigorously a Lyapunov exponent for three-dimensional quadratic mappings. Some elementary examples are also given and discussed.     

Travelling solitary wave solutions for the generalized Burgers--Huxley equation with nonlinear terms of any order

In this paper, the travelling wave solutions for the generalized Burgers--Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.     

A new explicit multisymplectic integrator for the Kawahara-type equation

We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.     

An action for a classical string,the equation of motion and group invariant classical solutions

A string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.      

Finite difference approximations for the fractional advection-diffusion equation

Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection-diffusion equation, the fractional Crank-Nicholson method (FCN) is given, which is based on shifted Grünwald-Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order ²(Δt). A numerical example is presented to confirm the conclusions.