Quasi-periodicity with respect to time of spatially periodic finite-gap solutions of the Camassa-Holm equation
We consider the spatially periodic problem for the recently derived dispersive shallow water equation of \textit{R. Camassa} and \textit{D. D. Holm} [Phys. Rev. Letters 71, 1661-1664 (1993)] \[ u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx},\tag{1} \] where \(u\) is the height of the water's free surface above a flat bottom. To analyze the initial value problem we follow Camassa-Holm by introducing a spectral problem in which the time \(t\) appears only as a parameter: \[ y''= \textstyle{{1\over 4}} y+\lambda my\tag{2} \] (here \(m= u- u_{xx}\)). We look for those values of \(\lambda\in \mathbb{R}\) for which equation (2) has a periodic or anti-periodic solution on \([0,1]\). While the solution \(u(t,x)\) of the Camassa-Holm equation develops in a complicated way, the periodic and anti-periodic spectrum remain unchanged. Consider (1) with \(u(0,x)\in C^3(\mathbb{R})\) of period 1 such that \(m(0,x)= u(0,x)- u_{xx}(0,x)\) gives rise to a finite simple spectrum. In this case, we proved that (1) has a unique solution \(u(t,x)\). This solution is spatially periodic (with period 1) and is defined for all values of \(t\geq 0\). Such solutions are called finite-gap situations. The aim of this note is to show that any smooth spatially periodic finite-gap solution to (1) is quasi-periodic with respect to time.
scientific article; zbMATH DE number 1228571