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. Parallels are drawn through discussion and example to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSo-lutions.m can be used to automatically compute traveling wave solutions of nonlin-ear 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 (supplied by the Publisher): Distribution format (supplied by the Program Library): 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 XP Programming language used: Mathematica Memory required to execute with typical data: 9 MB Number of processors used: 1 Has the code been vectorised or parallelized?: No Number of bytes in distributed program, including test data, etc.: 104 761 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 placed in a travel-ing frame of reference, the coefficients of a candidate polynomial solution in tanh are solved for. The resulting solution is 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 (such as Toda, Volterra, and Ablowitz-Ladik lattices) is 0.228 seconds with a standard deviation of 0.165 seconds on a 2.4GHz Pentium 4 with 512 MB RAM running Mathematica 4.1. The running time may vary considerably, depending on the complexity of the problem.