Willy Hereman

The tanh method: I. Exact solutions of nonlinear evolution and wave equations

A systemized version of the tanh method is used to solve particular evolution and wave equations. If one deals with conservative systems, one seeks travelling wave solutions in the form of a finite series in tanh. If present, boundary conditions are implemented in this expansion. The associated velocity can then be determined a priori, provided the solution vanishes at infinity. Hence, exact closed form solutions can be obtained easily in various cases.

The tanh method: II. Perturbation technique for conservative systems

With the aid of the tanh method, nonlinear wave equations are solved in a perturbative way. First, the KdVBurgers equation is investigated in the limit of weak dispersion. As a result, a general shock wave profile, with a perturbative solitary-wave contribution superposed, emerges. For a particular choice of the parameters, a comparison with the exact solution is made. Further, the MKdVBurgers is investigated in the same limit and similar results are obtained.

Symbolic methods to construct exact solutions of nonlinear partial differential equations

Two straightforward methods for finding solitary-wave and soliton solutions are presented and applied to a variety of nonlinear partial differential equations. The first method is a simplied version of Hirotas method. It is shown to be an effective tool to explicitly construct. multi-soliton solutions of completely integrable evolution equations of fifth-order, including the Kaup-Kupershmidt equation for which the soliton solutions were not previously known. The second technique is the truncated Painleve expansion method or singular manifold method. It is used to find closed-form solitary-wave solutions of the Fitzhugh-Nagumo equation with convection term, and an evolution equation due to Calogero. Since both methods are algorithmic, they can be implemented in the language of any symbolic manipulation program.

Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA

The direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized. The class of solitary wave solutions is extended to analytic (rather than rational) functions of the real exponential solutions of the linearized equation. Expanding the solutions in an infinite series in these real exponentials, an exact solution of the nonlinear PDE is obtained, whenever the series can be summed. Methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed. The algorithm is now suited to solving nonlinear equations by any symbolic manipulation program. This direct method is illustrated by constructing exact solutions of a generalized KdV equation, the Kuramoto-Sivashinski equation and a generalized Fisher equation.

Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method

The authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations. The physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system. The emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples it is shown how various cases of pulse-type and kink-type solutions are to be obtained by this method. An exhaustive list of equations so treated is presented in tabular form, together with the particular intermediate relations necessary for deriving their solutions. The extension of the technique to construct N-soliton solutions and indicate connections with other existing methods is outlined.

Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations

A new algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented. The algorithm is implemented inMathematica. The programcondens.mautomatically carries out the lengthy symbolic computations for the construction of conserved densities. The code is tested on several well-known partial differential equations from soliton theory. For systems with parameters,condens.mcan be used to determine the conditions on these parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for integrability of the system.

The computer calculation of Lie point symmetries of large systems of differential equations

Abstract A MACSYMA program is presented that greatly helps in the calculation of Lie symmetry groups of large systems of differential equations. The program calculates the determining equations for systems of m differential equations of order k , with p independent and q dependent variables, where m , k , p and q are arbitrary positive integers. The program automatically produces a list of determining equations for the coefficients of the vector field. This list has been parsed so that it is free of duplicate equations and trivial differential redundancies. Numerical factors and non-zero parameters occurring as factors are also removed. From the solution of these determining equations one can construct the Lie symmetry group. An example shows the use of the program in batch mode. It also illustrates a feedback mechanism, that not only allows the treatment of a large number of complicated partial differential equations but also aids in solving the determining equations step by step.

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

Abstract 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.

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

Abstract 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.m Catalogue identifier: ADUJ Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUJ Program obtainable from: CPC Program Library, Queens University of Belfast, N. Ireland Distribution format: tar.gz Computers: Created using a PC, but can be run on UNIX and Apple machines Operating systems under which the program has been tested: Windows 2000 and Windows XP Programming language used: Mathematica, version 3.0 or higher Memory required to execute with typical data: 9 MB Number of processors used: 1 Has the code been vectorised or parallelized?: No Number of lines in distributed program, including test data, etc.: 3221 Number of bytes in distributed program, including test data, etc.: 23 745 Nature 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.

Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA

Abstract A direct series method to find exact travelling wave solutions of nonlinear PDEs is applied to Hirotas system of coupled Korteweg-de Vries equations and to the sine-Gordon equation. The straightforward but lengthy algebraic computations to obtain single and multi-soliton solutions can be carried out with a symbolic manipulation package such as MACSYMA.