Automatic computation of the travelling wave solutions to nonlinear PDEs

Various extensions of the tanh-function method and their implementations for finding explicit travelling wave solutions to nonlinear partial differential equations (PDEs) have been reported in the literature. However, some solutions are often missed by these packages. In this paper, a new algorithm and its implementation called TWS for solving single nonlinear PDEs are presented. TWS is implemented in Maple 10. It turns out that, for PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases.

Program summary

Program title:TWSCatalogue identifier:AEAM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAM_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:1250No. of bytes in distributed program, including test data, etc.:78 101Distribution format:tar.gzProgramming language:Maple 10Computer:A laptop with 1.6 GHz Pentium CPUOperating system:Windows XP ProfessionalRAM:760 MbytesClassification:5Nature of problem:Finding the travelling wave solutions to single nonlinear PDEs.Solution method:Based on tanh-function method.Restrictions:The current version of this package can only deal with single autonomous PDEs or ODEs, not systems of PDEs or ODEs. However, the PDEs can have any finite number of independent space variables in addition to time t.Unusual features:For PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases.Additional comments:It is easy to use.Running time:Less than 20 seconds for most cases, between 20 to 100 seconds for some cases, over 100 seconds for few cases.References:[1] E.S. Cheb-Terrab, K. von Bulow, Comput. Phys. Comm. 90 (1995) 102.[2] S.A. Elwakil, S.K. El-Labany, M.A. Zahran, R. Sabry, Phys. Lett. A 299 (2002) 179.[3] E. Fan, Phys. Lett. 277 (2000) 212.[4] W. Malfliet, Amer. J. Phys. 60 (1992) 650.[5] W. Malfliet, W. Hereman, Phys. Scripta 54 (1996) 563.[6] E.J. Parkes, B.R. Duffy, Comput. Phys. Comm. 98 (1996) 288.     

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.     

Discrete tanh method for nonlinear difference-differential equations

By introducing a simple difference equation to deduce the difference terms and a simple differential equation to deduce the differential terms, we proposed an unified algebraic method for constructing exact solutions to difference-differential equations (DDEs). This method could give many kinds of exact solutions including soliton solutions expressed by hyperbolic functions, periodic solutions expressed by trigonometric functions and rational solutions in a uniform way if solutions of these kinds exist. In this paper, we also give a generalization of the method to determine the degree of DDEs, and compared with the creativity work of D. Baldwin et al. [D. Baldwin, Ü. Göktas, W. Hereman, Comput. Phys. Comm. 162 (2004) 203-217] through the discrete Hybrid equation.     

GeM software package for computation of symmetries and conservation laws of differential equations

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results.

Program summary

Title of program: GeMCatalogue identifier: ADYK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS SolarisOperating systems under which the program has been tested: Windows 2000, Windows XP, Linux, SolarisProgramming language used: Maple 9.5Memory required to execute with typical data: below 100 MegabytesNo. of lines in distributed program, including test data, etc.: 4939No. of bytes in distributed program, including test data, etc.: 166 906Distribution format: tar.gzNature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations.Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions.Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully.Typical running time: 1-20 seconds for problems that do not involve classification; 5-1000 seconds for problems that involve classification, depending on complexity.     

A Maple package for verifying ultradiscrete soliton solutions

We present a computer algebra program for verifying soliton solutions of ultradiscrete equations in which both dependent and independent variables take discrete values. The package is applicable to equations and solutions that include the max function. The program is implemented using Maple software.

Program summary

Program title: UltdeCatalogue identifier: AEDB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3171No. of bytes in distributed program, including test data, etc.: 13 633Distribution format: tar.gzProgramming language: Maple 10Computer: PC/AT compatible machineOperating system: Windows 2000, Windows XPRAM: Depends on the problem; minimum about 1 GBWord size: 32 bitsClassification: 5Nature of problem: The existence of multi-soliton solutions strongly suggest the integrability of nonlinear evolution equations. However enormous calculation is required to verify multi-soliton solutions of ultradiscrete equations. The use of computer algebra can be helpful in such calculations.Solution method: Simplification by using the properties of max-plus algebra.Restrictions: The program can only handle single ultradiscrete equations.Running time: Depends on the complexity of the equation and solution. For the examples included in the distribution the run times are as follows. (Core 2 Duo 3 GHz, Windows XP)

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Example 1: 2725 sec.

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Example 2: 33 sec.

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Example 3: 1 sec.

     

Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations

If a partial differential equation is reduced to an ordinary differential equation in the form u^′(ξ)=G(u,θ₁,…,θm) under the traveling wave transformation, where θ₁,…,θm are parameters, its solutions can be written as an integral form . Therefore, the key steps are to determine the parameters' scopes and to solve the corresponding integral. When G is related to a polynomial, a mathematical tool named complete discrimination system for polynomial is applied to this problem so that the parameter's scopes can be determined easily. The complete discrimination system for polynomial is a natural generalization of the discrimination △=b²−4ac of the second degree polynomial ax²+bx+c. For example, the complete discrimination system for the third degree polynomial F(w)=w³+d₂w²+d₁w+d₀ is given by and . In the paper, we give some new applications of the complete discrimination system for polynomial, that is, we give the classifications of traveling wave solutions to some nonlinear differential equations through solving the corresponding integrals. In finally, as a result, we give a partial answer to a problem on Fan's expansion method.     

Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations

With the aid of computerized symbolic computation, the extended Jacobian elliptic function expansion method and its algorithm are presented by using some relations among ten Jacobian elliptic functions and are very powerful to construct more new exact doubly-periodic solutions of nonlinear differential equations in mathematical physics. The new (2+1)-dimensional complex nonlinear evolution equations is chosen to illustrate our algorithm such that sixteen families of new doubly-periodic solutions are obtained. When the modulus m→1 or 0, these doubly-periodic solutions degenerate as solitonic solutions including bright solitons, dark solitons, new solitons as well as trigonometric function solutions.     

Classification of integrable super-systems using the SsTools environment

A classification problem is proposed for supersymmetric evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson-fermion systems are discovered under the weighting assumption . The syntax of the Reduce package SsTools, which was used for intermediate computations, and the applicability of its procedures to the calculus of super-PDE are described.

Program summary

Program title:SsToolsCatalogue identifier:ADYY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYY_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:89 178No. of bytes in distributed program, including test data, etc.:869 212Distribution format:tar.gzProgramming language:REDUCE 3.7, REDUCE 3.8Computer:(i) IBM PC, (ii) clusterOperating system:LINUXRAM:problem dependent (10 Mb-1 Gb), typical working size <100 MbWord size:32, 64 bitsNature of problem:The program allows the classification of N?1 supersymmetric nonlinear scaling-invariant evolution equations that admit infinitely many local symmetries propagated by recursion operators; here b(x,t;θ) is the set of bosonic super-fields and f(x;t;θ) are fermionic super-fields.Solution method:First, (half-)integer weights |f|,|b|,…,|Dt|,|Dx|≡1 are assigned to all variables and derivatives and then pairs of commuting flows that are homogeneous w.r.t. these weights are constructed. Secondly, the seeds of higher symmetry sequences [P.J. Olver, Applications of Lie Groups to Differential Equations, second ed., Springer, Berlin, 1993] for the systems are sorted out, and finally the recursion operators that generate the symmetries are obtained [I.S. Krasil'shchik, P.H.M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, Kluwer, Dordrecht, 2000]. The intermediate algebraic systems upon the undetermined coefficients are solved by using [T. Wolf, Applications of Crack in the classification of integrable systems, in: CRM Proc. Lecture Notes, vol. 37, 2004, pp. 283-300].Restrictions:Computation of symmetries of high differential order for very large evolutionary systems may cause memory restrictions. Additional size/time restrictions may occur if the homogeneity weights of some super-fields are non-positive, see Section 1.2Unusual features:SsTools has been extensively tested using hundreds of PDE systems within three years on UNIX-based PC-machines. SsTools is applicable to the computation of symmetries, conservation laws, and Hamiltonian structures for N?1 evolutionary super-systems with any N. SsTools is also useful for performing extensive arithmetic of general nature including differentiations of super-field expressions.Running time:Depends on the size and complexity of the input system and varies between seconds and minutes.References:[1] http://lie.math.brocku.ca/crack/susy/sstools.red; The support package Crack is obtained from http://lie.math.brocku.ca/crack/src/crack.tar.gz     

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.     

Adaptive Mesh Refinement for conservative systems: multi-dimensional efficiency evaluation

Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro- and magnetohydrodynamic simulations, AMR is used in combination with several shock-capturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multi-dimensional calculations and only 1.5-8% overhead is observed. For AMR calculations of multi-dimensional magnetohydrodynamic problems, several strategies for controlling the constraint are examined. Three source term approaches suitable for cell-centered representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing Lax-Friedrichs method on the coarser levels. This level-dependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.     

Mimetic discretization of two-dimensional Darcy convection

We consider discretization of the planar convection of the incompressible fluid in a porous medium filling rectangular enclosure. This problem belongs to the class of cosymmetric systems and admits an existence of a continuous family of steady states in the phase space. Mimetic finite-difference schemes for the primitive variables equation are developed. The connection of a derived staggered discretization with a finite-difference approach based on the stream function and temperature equations is established. Computations of continuous cosymmetric families of steady states are presented for the case of uniform and nonuniform grids.     

A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations

A direct approach with computerized symbolic computation is applied to construct a series of traveling wave solutions for nonlinear equations. Compared with most existing symbolic computation methods such as tanh method and Jacobi function method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solution according to some parameters.     

Staggered grids discretization in three-dimensional Darcy convection

We consider three-dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite-difference scheme for the Darcy convection problem in the primitive variables is developed. It consists of staggered nonuniform grids with five types of nodes, differencing and averaging operators on a two-nodes stencil. The nonlinear terms are approximated using special schemes. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. Branching off of a continuous family of steady states was detected for the problem with zero heat fluxes on two opposite lateral planes.     

Numerical realizations of solutions of the stochastic KdV equation

We investigate simulations of exact solutions of the stochastic Korteweg–deVries equation under additive noise. We compare the expectation values of the exact solutions to theoretical expectation values and to the numerical simulations of the stochastic Korteweg–deVries equation with and without damping. We find on average the diffused soliton vanishes long before the typically reported asymptotic limit.     

Fast LP method for the Schrödinger equation

The LP and CP methods are two versions of the piecewise perturbation methods to solve the Schrödinger equation. On each step the potential function is approximated by a constant (for CP) or by a linear function (for LP) and the deviation of the true potential from this approximation is treated by the perturbation theory.This paper is based on the idea that an LP algorithm can be made faster if expressed in a CP-like form. We obtain a version of order 12 whose two main ingredients are a new set of formulae for the computation of the zeroth-order solution which replaces the use of the Airy functions, and a convenient way of expressing the formulae for the perturbation corrections. Tests on a set of eigenvalue problems with a very big number of eigenvalues show that the proposed algorithm competes very well with a CP version of the same order and is by one order of magnitude faster than the LP algorithms existing in the literature. We also formulate a new technique for the step width adjustment and bring some new elements for a better understanding of the energy dependence of the error for the piecewise perturbation methods.     

A modified WTC algorithm for the Painlevé test of nonlinear variable-coefficient PDEs

A modified WTC algorithm for the Painlevé test of nonlinear PDEs with variable coefficients is proposed. Compared to the Kruskal's simplification algorithm, the modified algorithm further simplifies the computation in the third step of the Painlevé test for variable-coefficient PDEs to some extent. Two examples illustrate the proposed modified algorithm.     

Solution of the Schrödinger equation by a high order perturbation method based on a linear reference potential

The paper is devoted to the enhancement of the accuracy of the line-based perturbation method via the introduction of the perturbation corrections. We effectively construct the first and the second order corrections. We also perform the error analysis to predict that the introduction of successive corrections substantially enhances the order of the method from four, for the zeroth order version, to six and ten when the first and the second-order corrections are included. In order to remove the effect of the accuracy loss due to near-cancellation of like-terms when evaluating the perturbation corrections we construct alternative asymptotic formulae using a Maple code. We also propose a procedure for choosing the step size in terms of the preset accuracy and give a number of numerical illustrations.     

一类捕食-食饵系统的新的行波解

利用改进的[(G/G)-]展开法,借助软件的符号计算功能,求出了一类用来描述捕食-食饵群落时空动力性且食饵的平均生长率具有Allee效应的非线性偏微分方程组的新的行波解,这些解的性质合理地反映了生物入侵问题与参数值之间的相互依赖关系。     

A comparative study of some computer algebra packages which determine the Lie point symmetries of differential equations

The study seeks to determine which of five computer algebra packages is best at finding the Lie point symmetries of systems of partial differential equations with minimal user intervention. The chosen packages are LIEPDE and DIMSYM for REDUCE, LIE and BIGLIE for MUMATH, DESOLV for MAPLE, and MATHLIE for MATHEMATICA. A series of systems of partial differential equations are used in the study. The paper concludes that while all of the computer packages are useful, DESOLV appears to be the most successful system at determining the complete set of Lie point symmetries of systems of partial differential equations.     

Auto-Bäcklund transformation and exact solutions of the generalized variable-coefficient Kadomtsev-Petviashvili equation

Based on the idea of the homogeneous balance (HB) method, an auto-Bäcklund transformation (BT) to the generalized variable-coefficient Kadomtsev-Petviashvili (GvcKP) equation is obtained with symbolic computation. By the use of the auto-BT and the ε-expansion method, we can obtain a soliton-like solution including N-solitary wave of the GvcKP equation. Especially, we get a soliton-like solution including two-solitary wave as an illustrative example in detail. Since the cylindrical KP (cKP) equation, the generalized cKP (GcKP) equation and the spherical KP (SKP) equation are all special cases of the GvcKP equation, we can also obtain the corresponding results of these equations respectively.