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

The paper studies the Camassa-Holm (CH) equation

$\frac{\partial m}{\partial t}=-\left(mD+Dm\right)v,$

in which $D=\partial /\partial x$ and $m=v-{v}^{\text{'}\text{'}}$: in extenso,

$\text{CH}:\phantom{\rule{4pt}{0ex}}\frac{\partial v}{\partial t}-\frac{{\partial }^{3}v}{\partial t\partial {x}^{2}}+3v\frac{\partial v}{\partial x}-2\frac{\partial v}{\partial x}\frac{{\partial }^{2}v}{\partial {x}^{2}}-v\frac{{\partial }^{3}v}{\partial {x}^{3}}=0·$

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 $G={\left(1-{D}^{2}\right)}^{-1}=\frac{1}{2}{e}^{-|x-y|}$, the equation reads

${\text{CH}}^{\text{'}}:\phantom{\rule{4pt}{0ex}}\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}+\frac{\partial p}{\partial x}=0$

with the “pressure” $p=G\left[{v}^{2}+\frac{1}{2}{\left({v}^{\text{'}}\right)}^{2}\right]$. Tracking the moving “fluid” in the natural scale $\overline{x}=\overline{x}\left(t,x\right)$ determined by $\partial \overline{x}/\partial t=v\left(t,\overline{x}\right)$ with $\overline{x}\left(0,x\right)=x$, it reads

${\text{CH}}^{\text{'}\text{'}}:\phantom{\rule{4pt}{0ex}}\frac{d}{dt}v\left(t,\overline{x}\right)+{p}^{\text{'}}\left(\overline{x}\right)=0·$

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

##### MSC:
 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q35 PDEs in connection with fluid mechanics 37K15 Integration of completely integrable systems by inverse spectral and scattering methods 37K40 Soliton theory, asymptotic behavior of solutions 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction