Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2009-08-04
Nonlinear Sciences
Exactly Solvable and Integrable Systems
22 pages. To appear in a Special Issue of Applicable Analysis on Continuous and Discrete Integrable Systems (2009)
Scientific paper
Using standard calculus, explicit formulas for one-, two- and three-dimensional homotopy operators are presented. A derivation of the one-dimensional homotopy operator is given. A similar methodology can be used to derive the multi-dimensional versions. The calculus-based formulas for the homotopy operators are easy to implement in computer algebra systems such as Mathematica, Maple, and REDUCE. Several examples illustrate the use, scope, and limitations of the homotopy operators. The homotopy operator can be applied to the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs). Conservation laws provide insight into the physical and mathematical properties of the PDE. For instance, the existence of infinitely many conservation laws establishes the complete integrability of a nonlinear PDE.
Hereman Willy
Poole Douglas
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