Douglas Baldwin

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.

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order. A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved. The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.

Symbolic Software for the Painlevé Test of Nonlinear Ordinary and Partial Differential Equations

Abstract The automation of the traditional Painlevé test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of nonlinear ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants). Except where limited by memory, there is no restriction on the number of independent or dependent variables. The package is quite robust in determining all the possible dominant behaviors of the Laurent series solutions of the differential equation. The omission of valid dominant behaviors is a common problem in many implementations of the Painlevé test, and these omissions often lead to erroneous results. Finally, our package is compared with the other available implementations of the Painlevé test.

Nonlinear shallow ocean-wave soliton interactions on flat beaches.

Ocean waves are complex and often turbulent. While most ocean-wave interactions are essentially linear, sometimes two or more waves interact in a nonlinear way. For example, two or more waves can interact and yield waves that are much taller than the sum of the original wave heights. Most of these shallow-water nonlinear interactions look like an X or a Y or two connected Ys; at other times, several lines appear on each side of the interaction region. It was thought that such nonlinear interactions are rare events: they are not. Here we report that such nonlinear interactions occur every day, close to low tide, on two flat beaches that are about 2000 km apart. These interactions are closely related to the analytic, soliton solutions of a widely studied multidimensional nonlinear wave equation. On a much larger scale, tsunami waves can merge in similar ways.

Soliton generation and multiple phases in dispersive shock and rarefaction wave interaction

Interactions of dispersive shock waves (DSWs) and rarefaction waves (RWs) associated with the Korteweg-de Vries equation are shown to exhibit multiphase dynamics and isolated solitons. There are six canonical cases: one is the interaction of two DSWs that exhibit a transient two-phase solution but evolve to a single-phase DSW for large time; two tend to a DSW with either a small amplitude wave train or a finite number of solitons, which can be determined analytically; two tend to a RW with either a small wave train or a finite number of solitons; finally, one tends to a pure RW.

Interactions and asymptotics of dispersive shock waves – Korteweg–de Vries equation

Abstract The long-time asymptotic solution of the Korteweg–de Vries equation for general, step-like initial data is analyzed. Each sub-step in well-separated, multi-step data forms its own single dispersive shock wave (DSW); at intermediate times these DSWs interact and develop multiphase dynamics. Using the inverse scattering transform and matched-asymptotic analysis it is shown that the DSWs merge to form a single-phase DSW, which is the ‘largest’ one possible for the boundary data. This is similar to interacting viscous shock waves (VSW) that are modeled with Burgersʼ equation, where only the single, largest-possible VSW remains after a long time.