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On solitons, compactons, and Lagrange maps. (English) Zbl 1059.35524
Summary: Two local conservation laws of the K(m,n) equation u t ±(u m ) x +(u n ) xxx =0 are used to define two Lagrange-type transformations into mass and momentum space. These mappings help to identify new integrable cases (K(-1,-2), K(-2,-2), K(3 2,-1 2)), transform conventional solitary waves into compactons – solitary waves on a compactum – and relate certain soliton-carrying systems with compacton-carrying systems. Integrable equations are transformed into new integrable equations, and interaction of N-solitons of the m-KdV (m=3,n=1) is thus projected into an interaction in a compact domain from which N ordered stationary compactons emerge. The interaction of traveling compactons is the image of super-imposed equilibria of the corresponding soliton equation. For m=n+2, the potential form of the K(m,n) equation may also be cast into a conserved form and thus transformed, yielding generalized Dym and Wadati equations and two new integrable cases. It is shown that r t +(1-r 2 ) 3/2 (r xx +r) x =0 is integrable and supports compact kinks.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations