New travelling wave solutions to modified CH and DP equations

A new procedure for finding exact travelling wave solutions to the modified Camassa-Holm and Degasperis-Procesi equations is proposed. It turns out that many new solutions are obtained. Furthermore, these solutions are in general forms, and many known solutions to these two equations are only special cases of them.     

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.     

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.     

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.     

Searching for Painlevé integrable conditions of nonlinear PDEs with constant parameters using symbolic computation

Based on the Kruskal simplification for the WTC method, a generalized algorithm is devised to establish P-integrable (Painlevé integrable) conditions for nonlinear PDEs with multiple constant parameters. The generalized algorithm fully considers the impact of parameter coefficients upon every step of the Painlevé test. For parametric constraints obtained in the resonance analysis and verification of resonant conditions, the original equations should be regarded as a new system and repeating the test again. For illustration, we apply the generalized algorithm to coupled Schrödinger-Boussinesq equations. Based on the generalized algorithm, a Maple package SPIC is presented, which attributes to derive P-integrable conditions for given nonlinear PDEs with general forms. The higher order water wave equation and coupled KdV equations are selected to illustrate the effectiveness of our package. As a result, some P-integrable conditions are in agreement with the known results, in addition, several new P-integrable models are first given.     

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.

     

A note on the Painlevé test for nonlinear variable-coefficient PDEs

For a generalized nonlinear PDEs with variable coefficients, it is not Painlevé integrable unless the variable coefficients satisfy certain constraint conditions. In this note a generalized algorithm is proposed for the Painlevé test of nonlinear variable-coefficient PDEs. For the three steps of Painlevé test, i.e. leading order analysis, resonance determination and verification of resonant conditions, the analysis of parametric constraints is similar to those of nonlinear PDEs with constant coefficients given in my previous work. The main difference lies in the coefficients of Laurent series should have proper dependence according to the types of variable coefficients. By this generalized algorithm, several important nonlinear variable-coefficient PDEs, including KdV equation, mKdV equation, KP equation, NLS equation and higher-order NLS equation are studied and, in addition to rederiving all known P-integrable conditions, some new P-integrable models are obtained with the assistance of Maple.     

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.     

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.     

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.     

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.     

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     

Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the tension spline function and finite difference approximations. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of O(k²+k²h²+h²) and O(k²+k²h²+h⁴). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

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.     

Symbolic computation of the phoretic acceleration of convex particles suspended in a non-uniform gas

A package has been developed for calculating analytic expressions for forces and torques onto an arbitrarily shaped convex tracer (aerosol) particle small compared to the mean free path of the surrounding nonequilibrium gas. The package Phoretic allows to compute analytical (and also numerical) expressions for forces and torques stemming from elastic and diffusive scattering processes parameterized by an accommodation coefficient. The method is based on calculating half-sphere integral tensors of arbitrary rank and on integrating forces and torques acting on surface elements. The surrounding gas is completely specified by an arbitrarily shaped velocity distribution function. Accordingly, Phoretic requires two inputs: A particle (surface) geometry and a velocity distribution function. For example, the particle may be a cylinder with flat end caps, and the distribution function the one of Maxwell (isotropic) or Grad (13th moment approximation). The package reproduces analytic results for spheres which were available in the literature, and the ones for other geometries (cylinders, cuboids, ellipsoids) which were, however, only partially available (some works considered only elastic collisions, others temperature, or pressure, or only velocity gradients, etc.). In addition, Phoretic takes into account angular velocities which have been usually neglected and become relevant for non-spherical particles. The package is geared towards the implementation of dynamical equations for aerosol particles suspended in dilute or semidilute gases and as such helps to obtain concentration profiles and mobilities of aerosol particles depending on their shape (distribution) and environmental conditions.

Program summary

Title of program:PhoreticCatalogue identifier:ADYI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYI_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: Persons requesting the program must sign the standard CPC-non-profit use license (see license agreement printed in every issue)Computer for which the program is designed and others on which it has been tested: All platforms with a monitorOperating systems or monitors under which the program has been tested: Linux, Windows XP, Unix, Mac-OSProgram language used: Mathematica^®, version 5.2 or later. Phoretic makes use of the DiscreteMath‘Combinatorica’ Mathematica^® packageMemory required to execute with typical data: 10 MByteNo. of lines in distributed program, including test data, etc.: 22 410No. of bytes in distributed program, including test data, etc.: 114 657Distribution format:tar.gzNature of physical problem: Starting from a non-uniform velocity distribution function of a gas in terms of its moments, i.e. field variables, and field gradients such as temperature, pressure, or velocity field, the problem is to analytically calculate forces and torques acting onto arbitrarily shaped convex tracer (aerosol) particles small in size compared to the mean free path of the gas. The collision process is modeled as a superposition of elastic and diffusive scattering processes (parameterized by 0?α?1).Method of solution: We implemented the solution to this problem in the symbolic programming language Mathematica^®. The program allows to specify an arbitrary shape of the tracer particle and an arbitrary distribution function of the gas and returns symbolic or numerical expressions for forces and torques. The solution requires the calculation of half-sphere and base surface integrals and subsequent symbolic algebraic and tensorial manipulations.Restrictions on the complexity of the problem: Not known. In case the software cannot calculate surface integrals analytically it offers the possibility to proceed with a numerical evaluation of the corresponding terms.Typical running time: Typical running times mostly depend on the shape of the tracer particle. For all examples coming together with the software distribution run times are below 5 minutes on a modern single-processor platform.     

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.     

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.     

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.     

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.     

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.