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Fredholm determinants and the Camassa-Holm hierarchy. (English) Zbl 1047.37047

The paper studies the Camassa-Holm (CH) equation

m t=-(mD+Dm)v,

in which D=/x and m=v-v '' : in extenso,

CH:v t- 3 v tx 2 +3vv x-2v x 2 v x 2 -v 3 v x 3 =0·

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 G=(1-D 2 ) -1 =1 2e -|x-y| , the equation reads

CH ' :v t+vv x+p x=0

with the “pressure” p=G[v 2 +1 2(v ' ) 2 ]. Tracking the moving “fluid” in the natural scale x ¯=x ¯(t,x) determined by x ¯/t=v(t,x ¯) with x ¯(0,x)=x, it reads

CH '' :d dtv(t,x ¯)+p ' (x ¯)=0·

The author proves that if m is summable at the start together with m ' , then the Lagrangian version CH '' of the CH flow is perfectly fine for all time 0t<. Moreover, the author integrates the CH '' equation explicitly in terms of certain theta-like Fredholm determinants, thereby providing expressions of v(t,x ¯) and the scale x ¯(t,x) that are always sensible for any (t,x)[0,)×. It is only v ' (t,x ¯) that misbehaves, and this does not spoil the Lagrangian version CH '' . Consequently, the Eulerian version CH ' is also fine, although v ' (t,x) can be infinite now and then. Finally, the paper also discusses some open questions concerning soliton trains and generalizations of the CH equation.

37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q35PDEs in connection with fluid mechanics
37K15Integration of completely integrable systems by inverse spectral and scattering methods
37K40Soliton theory, asymptotic behavior of solutions
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction