A second order numerical scheme for the improved Boussinesq equation

A finite-difference scheme arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation is used for the numerical solution of the improved Boussinesq equation (IBq). The resulting linear scheme, which is analyzed for local truncation error and stability, is tested numerically and conclusions with corresponding results known in the bibliography are derived.     

Phase-space moment-equation model of highly relativistic electron-beams in plasma-wakefield accelerators

We formulate a new procedure for modelling the transverse dynamics of relativistic electron beams with significant energy spread when injected into plasma-based accelerators operated in the blow-out regime. Quantities of physical interest, such as the emittance, are furnished directly from solution of phase space moment equations formed from the relativistic Vlasov equation. The moment equations are closed by an Ansatz, and solved analytically for prescribed wakefields. The accuracy of the analytic formulas is established by benchmarking against the results of a semi-analytic/numerical procedure which is described within the scope of this work, and results from a simulation with the 3D quasi-static PIC code HiPACE.     

Newton-conjugate-gradient methods for solitary wave computations

In this paper, the Newton-conjugate-gradient methods are developed for solitary wave computations. These methods are based on Newton iterations, coupled with conjugate-gradient iterations to solve the resulting linear Newton-correction equation. When the linearization operator is self-adjoint, the preconditioned conjugate-gradient method is proposed to solve this linear equation. If the linearization operator is non-self-adjoint, the preconditioned biconjugate-gradient method is proposed to solve the linear equation. The resulting methods are applied to compute both the ground states and excited states in a large number of physical systems such as the two-dimensional NLS equations with and without periodic potentials, the fifth-order KdV equation, and the fifth-order KP equation. Numerical results show that these proposed methods are faster than the other leading numerical methods, often by orders of magnitude. In addition, these methods are very robust and always converge in all the examples being tested. Furthermore, they are very easy to implement. It is also shown that the nonlinear conjugate gradient methods are not robust and inferior to the proposed methods.     

Analytic study on solitons in the nonlinear fibers with time-modulated parabolic law nonlinearity and Raman effect

In this paper, the nonlinear optical transmission equation with time-modulated cubic–quintic nonlinearity and Raman effect term, describing the ultrashort optical pulse propagate along the nonlinear media, has been studied by using the non-auto-Bäcklund transformation (NATB) and subsidiary ordinary differential equation (sub-ODE) expansion method. As a result, analytical solution expressions to that equation are obtained. Finally, two simple examples, including systems with the external potential in exponential form and periodic form, are provided to show the solution steps.     

A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows

We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) [21]. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor [14]. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.     

Adiabatic perturbations for compactons under dissipation and numerically-induced dissipation

Compacton propagation under dissipation shows amplitude damping and the generation of tails. The numerical simulation of compactons by means of dissipative schemes also show the same behaviors. The truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis. For dissipative schemes, or when artificial dissipation is added, the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically-induced tails. In this paper, such methods are applied to the K(2,2)$K(2\text{,}2)$ Rosenau–Hyman equation, showing a very good agreement between perturbative and numerical results.     

1-Soliton solution of the generalized Zakharov-Kuznetsov equation with nonlinear dispersion and time-dependent coefficients

In this Letter, the 1-soliton solution of the Zakharov-Kuznetsov equation with power law nonlinearity and nonlinear dispersion along with time-dependent coefficients is obtained. There are two models for this kind of an equation that are studied. The constraint relation between these time-dependent coefficients is established for the solitons to exist. Subsequently, this equation is again analysed with generalized evolution. The solitary wave ansatz is used to carry out this investigation.     

The general solution of two-dimensional matrix Toda chain equations with fixed ends

It is shown that the two-dimensional matrix Toda chain determines the group of discrete symmetries of the two-dimensional matrix nonlinear Schrödinger equation (the matrix generalization of the Davey-Stewartson system). The general solution of this chain with definite boundary conditions is obtained in explicit form.     

Numerical solution of the Fokker-Planck equation with variable coefficients

We study the numerical solution of the Fokker-Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.     

Cohomological uniqueness of the Cauchy problem solutions for the Einstein equation

In this paper we analyze the Cauchy problem for the Einstein equation in the case of a non-characteristic initial hypersurface. To find the correct notions for the characteristic and Cauchy data we introduce a complex, which we call the Einstein complex. Then the Cauchy problem acquires correctness in terms of the associated spectral sequence. We define the Cauchy data in such way that they allow us to reconstruct a cohomologously unique formal solution.     

An explicit one-soliton solution of the (2+1)-dimensional generalized sine-gordon equations

The (n+1)-dimensional differential geometric generalization of the sine-Gordon equation (SGE) given by Tenenblat and Terng is solved explicitly in the casen=2 to obtain a one-soliton solution. The solution yields the soliton solution of the (1+1)-dimensional SGE in the limit as one of the three independent variables approaches infinity. However, more than one variable plays the role of time in these limits.     

Investigation of oscillations of a soliton of a bose condensate of atoms in a trap in the limit of small oscillations of its walls

The motion of the center of a soliton in a trap with oscillating walls is studied analytically and numerically for the case in which the intrinsic frequency of small soliton oscillations in the equilibrium state considerably exceeds the frequency of wall oscillations. this problem can be solved either by applying the gross–pitaevskii equation, which most exactly describes the behavior of the soliton in the trap, or by using the approximate, “mechanical,” equation of motion of the newtonian type for the center of the soliton. an approximate analytical solution of the mechanical equation is obtained and is compared with the numerical solution of the newton equation, while the latter solution is compared with the numerical solution of the gross–pitaevskii equation. good agreement between the first two solutions is revealed. it is also shown that there is a range of parameters in which the numerical solutions of the newton and gross–pitaevskii equations are closest to each other. the frequency-sweeping effect of soliton center oscillations is revealed. an approximate analytical formula for the limiting frequency of these oscillations is obtained and the numerical analysis of this phenomenon is performed.     

Fast sweeping method for the factored eikonal equation

We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss–Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss–Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.     

Exact numerical solution for the diffused channel waveguide

We present an exact numerical solution based on a vector integral equation to investigate the diffused channel waveguide. Various examples are given and compared with other numerical and approximate methods. An approximate numerical solution is also given using the effective index method and the integral equation solution for the inhomogeneous slab.     

非线性中子输运问题的一个解法

本文推导了考虑中子之间碰撞的非线性中子输运方程,提出了求解该方程的n-n逐次碰撞展开法,并利用此方法把方程化为适于数值求解的线性输运方程组的形式,编制计算程序进行了数值计算,讨论了解的物理意义。     

The wave equation with viscoelastic attenuation and its application in problems of shallow-sea acoustics

A suitable tool for the simulation of low frequency acoustic pulse signals propagating in a shallow sea is the numerical integration of the nonstationary wave equation. The main feature of such simulation problems is that in this case the sound waves propagate in the geoacoustic waveguide formed by the upper layers of the bottom and the water column. By this reason, the correct dependence of the attenuation of sound waves in the bottom on their frequency must be taken into account. In this paper we obtain an integro-differential equation for the sound waves in the viscoelastic fluid, which allows to simulate the arbitrary dependence of acoustic wave attenuation on frequency in the time domain computations. The procedure of numerical solution of this equation based on its approximation by a system of differential equations is then considered and the methods of artificial limitation of computational domain are described. We also construct a simple finite-difference scheme for the proposed equation suitable for the numerical solution of nonstationary problems arising in the shallow-sea acoustics.     

Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation

In this Letter we propose a pseudospectral method for solving Thomas-Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on rational Chebyshev pseudospectral method. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.     

Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation

In this paper, we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation. In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation. The power of this manageable method is confirmed. Moreover, the use of Legendre wavelet is found to be accurate, simple and fast.     

Moving mesh methods for blowup in reaction–diffusion equations with traveling heat source

In this paper moving mesh methods are used to simulate the blowup in a reaction–diffusion equation with traveling heat source. The finite-time blowup occurs if the speed of the movement of the heat source remains sufficiently low, and the blowup procedure is not fixed at one point not like that for stationary heat source. As time goes to the blowup time, the blowup profile converges to a stationary state. In the simulation a new moving mesh algorithm is designed to deal with the difficulty caused by the delta function in the traveling heat source. The convergence rates are verified and new blowup figures are generated from the numerical experiments.     

关于干涉反演光谱技术能否扩大F-P干涉仪光谱范围的探讨

本文从物理学的角度提出了Fredholm第一类积分方程数值解的可靠性概念,证明了在被称为Fabry-Perot干涉反演光谱技术中,当△σ=2/x,△σ=2/σ时,若取样点数为一个适当的奇数,那么积分方程的数值解是稳定的.但是进一步的计算机模拟实验表明,该数值解不是原积分方程的可靠解,因此,干涉反演光谱技术不能扩大Fabry-Perot干涉仪的光谱范围.理论分析表明,在前述条件下,积分方程数值解不可靠的根本原因在于该积分方程本身没有唯一解.