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.     

Hidden solitons in the Zabusky–Kruskal experiment: Analysis using the periodic,inverse scattering transform

Recent numerical work on the Zabusky–Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne’s nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corroborated, and the exact number of solitons, their amplitudes and their reference level is computed. Other “less nonlinear” oscillation modes, which are not solitons, are also found to have nontrivial energy contributions over certain ranges of the dispersion parameter. In addition, the reference level is found to be a non-monotone function of the dispersion parameter. Finally, in the case of large dispersion, we show that the one-term nonlinear Fourier series yields a very accurate approximate solution in terms of Jacobian elliptic functions.     

Modified nonlinearly dispersive mK(m,n,k) equations: II. Jacobi elliptic function solutions

Recently we have obtained compacton solutions and solitary pattern solutions of the modified nonlinearly dispersive KdV equations (simply called mK(m,n,k) equations). In this paper the mK(m,n,k) equations are investigated again. By using some transformations we give their some Jacobi elliptic function solutions. When the modulus μ→1 or 0, some of the obtained Jacobi elliptic function solutions degenerate as solitary wave solution.     

非线性演化方程的新Jacobi椭圆函数解

基于sinh-Gordon方程的椭圆函数解,构造新的试探解来扩展sinh-Gordon方程展开法.利用该方法研究了KdV-mKdV方程,双sine-Gordon方程和BBM方程,获得了这些方程的新Jacobi椭圆函数解.该方法也能用来求解其他数学物理中的非线性演化方程.     

RAEEM: A Maple package for finding a series of exact traveling wave solutions for nonlinear evolution equations

In Maple 8, by taking advantage of the package RIF contained in DEtools, we developed a package RAEEM which is a comprehensive and complete implementation of such methods as the tanh-method, the extended tanh-method, the Jacobi elliptic function method and the elliptic equation method. RAEEM can entirely automatically output a series of exact traveling wave solutions, including those of polynomial, exponential, triangular, hyperbolic, rational, Jacobi elliptic, Weierstrass elliptic type. The effectiveness of the package is illustrated by applying it to a large variety of equations. In addition to recovering previously known solutions, we also obtain more general forms of some solutions and new solutions.

Program summary

Title of program: RAEEMCatalogue identifier: ADUPProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUPProgram obtained from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: PC Pentium IVInstallations: CopyOperating systems: Windows 98/2000/XPProgram language used: Maple 8Memory required to execute with typical data: depends on the problem, minimum about 8M wordsNo. of bits in a word: 8No. of lines in distributed program, including test data, etc.: 3163No. of bytes in distributed program, including the test data, etc.: 26 720Distribution format: tar.gzNature of physical problem: Our program provides exact traveling wave solutions, which describe various phenomena in nature, and thus can give more insight into the physical aspects of problems. These solutions may be easily used in further applications.Restriction on the complexity of the problem: The program can handle system of nonlinear evolution equations with any number of independent and dependent variables, in which each equation is a polynomial (or can be converted to a polynomial) in the dependent variables and their derivatives.Typical running time: It depends on the input equations as well as the degrees of the desired polynomial solutions. For most of the equations we have computed, the running time is no more than 100 s.     

Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme

This paper proposes a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, as it arises, for instance, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation allows for soliton-type solutions, a ubiquitous phenomenon in a large variety of physical problems. The semidiscretization approach first leads to a system of second-order nonlinear ordinary differential equations. The system is then approximated by a nonlinear recurrence relation which involves a cosine function. The numerical solution of the system is obtained via a further application of a sequential splitting in a linearly implicit manner that avoids solving the nonlinear scheme at each time step and allows an efficient implementation of the simulation in a locally one-dimensional fashion. The new method has potential applications in further multi-dimensional nonlinear wave simulations. Rigorous analysis is given for the numerical stability. Numerical demonstrations for colliding circular solitons are given.     

An inverse convergence approach for arguments of Jacobian elliptic functions

Computing the value of the Jacobian elliptic functions, given the argument u and the parameter m, is a problem, whose solution can be found either tabulated in tables of elliptic functions [1] or by use of existing software, such as Mathematica, etc. The inverse problem, finding the argument, given the Jacobian elliptic function and the parameter m, is a problem whose solution is found only in tables of elliptic functions. Standard polynomial inverse interpolation procedures fail, due to ill conditioning of the system of the unknowns. In this paper, we describe a numerical procedure based on the convergence of the unknowns of the solution, by the use of arithmetical method, as an alternative way of solving the problem. The method gives very good results with no significant error, in the computation of the argument of the Jacobian elliptic function given the Jacobian elliptic function and the parameter. This new procedure is important in problems involving cavities or inclusions of ellipsoidal shape encountered in the mechanical design of bearings, filters, and composite materials. They are also important in the modeling of porosity of bones. This porosity may lead to osteoporosis, a disease which affects bone mineral density in humans with bad consequences. Also these procedures are of importance in problems encountered in the physics discipline such as in the analysis of the dependence of the maximum tunneling current on external magnetic field for large area Josephson junctions with overlap boundary conditions.     

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Through discussion and example, parallels are drawn to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.

Program summary

Title of program: DDESpecialSolutions.mCatalogue identifier:ADUJProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUJProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: Created using a PC, but can be run on UNIX and Apple machinesOperating systems under which the program has been tested: Windows 2000 and Windows XPProgramming language used: Mathematica, version 3.0 or higherMemory required to execute with typical data: 9 MBNumber of processors used: 1Has the code been vectorised or parallelized?: NoNumber of lines in distributed program, including test data, etc.: 3221Number of bytes in distributed program, including test data, etc.: 23 745Nature of physical problem: The program computes exact solutions to differential-difference equations in terms of the tanh function. Such solutions describe particle vibrations in lattices, currents in electrical networks, pulses in biological chains, etc.Method of solution: After the differential-difference equation is put in a traveling frame of reference, the coefficients of a candidate polynomial solution in tanh are solved for. The resulting traveling wave solutions are tested by substitution into the original differential-difference equation.Restrictions on the complexity of the program: The system of differential-difference equations must be polynomial. Solutions are polynomial in tanh.Typical running time: The average run time of 16 cases (including the Toda, Volterra, and Ablowitz-Ladik lattices) is 0.228 seconds with a standard deviation of 0.165 seconds on a 2.4 GHz Pentium 4 with 512 MB RAM running Mathematica 4.1. The running time may vary considerably, depending on the complexity of the problem.     

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi’s elliptic functions.For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi’s sn or cn functions. Examples illustrate key steps of the algorithms.The new algorithms are implemented in Mathematica. The package PDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed.A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.     

A quadratically convergent method for computing simple singular roots and its application to determining simple bifurcation points

We present an efficient method for computing roots of mappings on ? ⁿ in the case where the Jacobian has the rankn?1 at the root. For the accurate determination of such a rootx*∈? ⁿ an auxiliary system ofn equations inn+1 variables is constructed which possesses (x ^*, 1) as a turning point. This turning point can be computed by direct methods. We use an adapted method which requires only the solution of (n+1)-dimensional systems of linear equations and the evaluation of one Jacobian and 5 function values per step. This techniques is successfully applied to compute simple bifurcation points by means of a suitable system of nonlinear equations which has the properties mentioned above.     

A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains

We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.     

Efficient unsteady high Reynolds number flow computations on unstructured grids

Despite the advances in computer power and numerical algorithms over the last decades, solutions to unsteady flow problems remain computing time intensive. Especially for high Reynolds number flows, nonlinear multigrid, which is commonly used to solve the nonlinear systems of equations, converges slowly. The stiffness induced by the high-aspect ratio cells and turbulence is not tackled well by this solution method.In this paper, it is investigated if a Jacobian-free Newton-Krylov (jfnk) solution method can speed up unsteady flow computations at high Reynolds numbers. Preconditioning of the linear systems that arise after Newton linearization is commonly performed with matrix-free preconditioners or approximate factorizations based on crude approximations of the Jacobian. Approximate factorizations based on a Jacobian that matches the target residual operator are unpopular because these preconditioners consume a large amount of memory and can suffer from robustness issues. However, these preconditioners remain appealing because they closely resemble A^-1.In this paper, it is shown that a jfnk solution method with an approximate factorization preconditioner based on a Jacobian that approximately matches the target residual operator enables a speed up of a factor 2.5-12 over nonlinear multigrid for two-dimensional high Reynolds number flows. The solution method performs equally well as nonlinear multigrid for three-dimensional laminar problems. A modest memory consumption is achieved with partly lumping the Jacobian before constructing the approximate factorization preconditioner, whereas robustness is ensured with enhanced diagonal dominance.     

A sixth order fast direct helmholtz equation solver

An O(h⁶) accurate difference approximation to solutions of the Helmholtz equation is derived. The discrete equations are solved using a reduction procedure and Fourier analysis. Its computational performance is compared with a fourth order similar method over a set of linear and mildly nonlinear elliptic boundary value problems.     

Collision dynamics of elliptically polarized solitons in Coupled Nonlinear Schrödinger Equations

We investigate numerically the collision dynamics of elliptically polarized solitons of the System of Coupled Nonlinear Schrödinger Equations (SCNLSE) for various different initial polarizations and phases. General initial elliptic polarizations (not sech$sech$-shape) include as particular cases the circular and linear polarizations. The elliptically polarized solitons are computed by a separate numerical algorithm. We find that, depending on the initial phases of the solitons, the polarizations of the system of solitons after the collision change, even for trivial cross-modulation. This sets the limits of practical validity of the celebrated Manakov solution. For general nontrivial cross-modulation, a jump in the polarization angles of the solitons takes place after the collision (‘polarization shock’). We study in detail the effect of the initial phases of the solitons and uncover different scenarios of the quasi-particle behavior of the solution. In majority of cases the solitons survive the interaction preserving approximately their phase speeds and the main effect is the change of polarization. However, in some intervals for the initial phase difference, the interaction is ostensibly inelastic: either one of the solitons virtually disappears, or additional solitons are born after the interaction. This outlines the role of the phase, which has not been extensively investigated in the literature until now.     

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial.     

Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine–cosine method

In this paper, we establish exact solutions for coupled nonlinear evolution equations. The sine–cosine method is used to construct exact periodic and soliton solutions of coupled nonlinear evolution equations. Many new families of exact travelling wave solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equations and the coupled nonlinear Klein–Gordon and Nizhnik–Novikov–Veselov equations are successfully obtained. The obtained solutions include compactons, solitons, solitary patterns and periodic solutions. These solutions may be important and of significance for the explanation of some practical physical problems.     

A Maple package for finding exact solitary wave solutions of coupled nonlinear evolution equations

The tanh function expansion method for finding traveling solitary wave solutions to coupled nonlinear evolution equations is described. A complete implementation RATHS written in Maple is presented, in which the operator mains can output exact solitary wave solutions entirely automatically. Furthermore, RATHS can handle any number of dependent variables u_i as well as any number of independent variables x_j contained in the input system. This package can also be applied to ODEs. The effectiveness of RATHS is illustrated by applying it to a variety of equations.

Program summary

Title of program: RATHSCatalogue identifier: ADSD (also ADQK)Program Summary URL:http://cpc.cs.qub.ac.uk/summaries/ADRY (also ADQR)Program obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputers: PC Pentium IVInstallations: CopyOperating systems: Windows 98/2000/XPProgram language used: Maple V R6Memory required to execute with typical data: depends on the problem, minimum about 8M wordsNo. of bits in a word: 8No. of bytes in distributed program, including the test data, etc.: 16 608Distribution format:tar gzip fileKeywords: Coupled nonlinear evolution equations, traveling solitary wave solutions, dependent variable, independent variableNature of physical problem: Our program give out exact solitary wave solutions, which can describe various phenomena in nature, and thus can give more insight into the physical aspects of problems and may be easily used in further applications.Restriction on the complexity of the problem: The program can handle coupled nonlinear evolution equations, in which every equation is a polynomial (or can be converted to a polynomial) in the unknown functions and their derivatives.Typical running time: It depends on the input equations as well as the degrees of the desired polynomial solutions. For most of the coupled equations which we have computed, the running time is no more than 20 seconds.     

The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation

In this paper, the $N$-soliton solution is constructed for the ($2+1$)-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons and their interactions are obtained by choosing special parameters. Especially, by selecting appropriate parameters on the multi-soliton solutions, the two soliton can reduce to a periodic soliton or a lump soliton, the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton, while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons. Detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters. Finally, an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method, and the relevant interaction and propagation characteristics are discussed graphically. The results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two-dimensional nonlinear wave equations.     

New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations

The special exact solutions of nonlinearly dispersive Boussinesq equations (called B(m,n) equations), u_tt−u_xx−a(uⁿ)_xx+b(u^m)_xxxx=0, is investigated by using four direct ansatze. As a result, abundant new compactons: solitons with the absence of infinite wings, solitary patterns solutions having infinite slopes or cups, solitary waves and singular periodic wave solutions of these two equations are obtained. The variant is extended to include linear dispersion to support compactons and solitary patterns in the linearly dispersive Boussinesq equations with m=1. Moreover, another new compacton solution of the special case, B(2,2) equation, is also found.     

Domain wall arrays, fronts, and bright and dark solitons in a generalized derivative nonlinear Schrödinger equation

We obtain some exact solutions of a generalized derivative nonlinear Schrödinger equation, including domain wall arrays (periodic solutions in terms of elliptic functions), fronts, and bright and dark solitons. In certain parameter domains, fundamental bright and dark solitons are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. Moreover, we also find the chirping reversal phenomena of fronts, and bright and dark solitons, and discuss two different ways to produce the chirping reversal.