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Shallow Water Wave Systems
Author(s) -
Clarkson Peter A.,
Priestley Thomas J.
Publication year - 1998
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00099
Subject(s) - mathematics , symmetry (geometry) , mathematical physics , reduction (mathematics) , mathematical analysis , pure mathematics , physics , geometry
In this article we study various systems that represent the shallow water wave equation v xxt + αvv t − βv x ∂ x ‐1 ( v t ) − v t − v x = 0, where (∂ x −1 f )( x )=∫ x ∞ f ( y ) d y , and α and β are arbitrary, nonzero, constants. The classical method of Lie, the nonclassical method of Bluman and Cole [ J. Math. Mech. 18:1025 (1969)], and the direct method of Clarkson and Kruskal [ J. Math. Phys. 30:2201 (1989)] are each applied to these systems to obtain their symmetry reductions. It is shown that for both the nonclassical and direct methods unusual phenomena can occur, which leads us to question the relationship between these methods for systems of equations. In particular an example is exhibited in which the direct method obtains a reduction that the nonclassical method does not.