The paper studies the Camassa-Holm (CH) equation
in which and : in extenso,
The CH equation arises in the study of long waves in shallow water [R. Camassa and D. D. Holm, Phys. Rev. Lett. 71, 1661–1664 (1993; Zbl 0972.35521)] and its integrable properties are presented in soliton theory [B. Fuchssteiner, Physica. D 95, 229–243 (1996; Zbl 0900.35345)].
In terms of Green’s function , the equation reads
with the “pressure” . Tracking the moving “fluid” in the natural scale determined by with , it reads
The author proves that if is summable at the start together with , then the Lagrangian version CH of the CH flow is perfectly fine for all time . Moreover, the author integrates the CH equation explicitly in terms of certain theta-like Fredholm determinants, thereby providing expressions of and the scale that are always sensible for any . It is only that misbehaves, and this does not spoil the Lagrangian version CH. Consequently, the Eulerian version CH is also fine, although can be infinite now and then. Finally, the paper also discusses some open questions concerning soliton trains and generalizations of the CH equation.