Exterior Differential Systems Prolongations and Application to a Study of Two Nonlinear Partial Differential Equations

• Published in: Acta applicandae mathematicae Vol. 113; no. 3; pp. 247 - 263
• Main Author: Bracken, Paul
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2011; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Conservation laws; Differential equations; Exteriors; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear differential equations; Nonlinear equations; Partial Differential Equations; Probability Theory and Stochastic Processes; Prolongation; Studies; Transformations; 37J35; 37K35; 37K25; 53Z05; 35G20; Differential system; Prolongation structure; Integrability; Camassa-Holm; 35Q53; Korteweg-de Vries; Nonlinear equation; 55R10
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-010-9597-z

A generalized Korteweg-de Vries equation is formulated as an exterior differential system, which is used to determine the prolongation structure of the equation. The prolongation structure is obtained for several cases of the variable powers, and nontrivial algebras are determined. The analysis is extended to a differential system which gives the Camassa-Holm equation as a particular case. The subject of conservation laws is briefly discussed for each of the equations. A Bäcklund transformation is determined using one of the prolongations.