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On the scattering problem for the Camassa-Holm equation. (English) Zbl 0999.35065
From the introduction: The paper is devoted to the scattering problem for the nonlinear partial differential equation \[ \begin{alignedat}{2} u_t-u_{txx}+2\omega u_x+3uu_x &= 2u_xu_{xx}+u_{xxx},\quad &&t>0, \;x\in\mathbb{R},\\ u(0,x) &= u_0(x),\quad &&x\in\mathbb{R}. \end{alignedat}\tag{1.1} \] With \(m := u - u_{xx}\), called the potential (physically it represents the momentum) equation (1.1) can be expressed as the condition of compatibility between \[ \psi''=\frac 14 \psi+\lambda m\psi+\lambda\omega\psi\tag{1.2} \] and \[ \partial_t\psi=\left(\frac{1}{2\lambda}-u\right)\psi'+\frac 12 u'\psi,\tag{1.3} \] that is, \[ \partial_t(\psi'')=(\partial_t\psi)'' \] is the same as to say that (1.1) holds. Equation (1.2) is the spectral problem associated to (1.1). All spectral characteristics of (1.2) are constants of motion under the Camassa-Holm flow. Equation (1.1) can be considered in the class of spatially periodic functions or in the class of functions on the line decaying at infinity. Accordingly, the isospectral problem (1.2) is a periodic weighted Sturm-Liouville problem or a weighted spectral problem in \(L^2(\mathbb{R})\). In §2 we give an exact description of the spectrum of (1.2) in \(L^2(\mathbb{R})\) under the assumption that \(m\in H^1(\mathbb{R})\) satisfies \(\int_{\mathbb{R}}(1+|x|)|m(x)|dx<\infty\). After a discussion of the initial value problem (1.1), we proceed in §3 with the determination of the evolution under the Camassa-Holm flow of the scattering data associated to an initial profile for (1.1) in the absence of bound states. We solve the inverse scattering problem for a special class of scattering data.

MSC:
35P25 Scattering theory for PDEs
35Q35 PDEs in connection with fluid mechanics
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