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Zbl 1047.37047
McKean, Henry P.
Fredholm determinants and the Camassa-Holm hierarchy. (English)
[J] Commun. Pure Appl. Math. 56, No. 5, 638-680 (2003). ISSN 0010-3640

The paper studies the Camassa-Holm (CH) equation $$ \frac {\partial m}{\partial t}=-(mD+Dm)v,$$ in which $ D= \partial /{\partial x}$ and $ m=v-v''$: in extenso, $$ \text{CH}:\ \frac {\partial v}{\partial t}-\frac {\partial ^3v}{\partial t\partial x^2}+3v\frac {\partial v}{\partial x}-2\frac {\partial v}{\partial x}\frac {\partial ^2v}{\partial x^2}-v\frac {\partial ^3 v}{\partial x^3}=0. $$ The CH equation arises in the study of long waves in shallow water [{\it R. Camassa} and {\it D. D. Holm}, Phys. Rev. Lett. 71, 1661--1664 (1993; Zbl 0972.35521)] and its integrable properties are presented in soliton theory [{\it B. Fuchssteiner}, Physica. D 95, 229--243 (1996; Zbl 0900.35345)]. \par In terms of Green's function $G=(1-D^2)^{-1}=\frac 12e^{-\vert x-y\vert }$, the equation reads $$ \text{CH}':\ \frac {\partial v}{\partial t}+v\frac {\partial v}{\partial x}+\frac {\partial p}{\partial x}=0 $$ with the ``pressure" $p=G[v^2+\frac 12(v')^2]$. Tracking the moving ``fluid" in the natural scale $\bar x=\bar x (t,x)$ determined by $\partial \bar x /\partial t=v(t,\bar x)$ with $\bar x(0,x)=x$, it reads $$ \text{CH}'':\ \frac {d }{d t}v(t,\bar x)+p'(\bar x)=0 .$$ 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 $0\le t<\infty$. Moreover, the author integrates the CH$''$ equation explicitly in terms of certain theta-like Fredholm determinants, thereby providing expressions of $v(t,\bar x)$ and the scale $\bar x(t,x)$ that are always sensible for any $(t,x)\in [0,\infty)\times \Bbb{R}$. It is only $v'(t,\bar x)$ 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.
[Ma Wen-Xiu (Tampa)]

MSC 2000:: *37K10 Completely integrable systems etc.
35Q35 Other equations arising in fluid mechanics
37K15 Integration by inverse spectral and scattering methods
37K40 Soliton theory, asymptotic behavior of solutions
76B15 Wave motions (fluid mechanics)

Keywords: Camassa-Holm hierarchy; Fredholm determinants; soliton theory; long waves in shallow water; integrable properties

Citations: Zbl 0972.35521; Zbl 0900.35345

Cited in: Zbl 1157.35484

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