Well-posedness of modified Camassa-Holm equations
This paper studies the Cauchy problem for the class of modified Camassa-Holm (mCH) equations given by \[ (\partial_t+u\partial_x+2u_x)m=0, \qquad m=(I-\partial_x^2)^ku, \tag{1} \] where \(k\geq2\) is a positive integer. The case \(k=1\) corresponds to the now celebrated Camassa-Holm (CH) equation. The authors begin by deriving Eq.~(1) as a geodesic equation on some diffeomorphism group with respect to the invariant \(H^k\) metric. Using Kato theory, they establish local well-posedness for the mCH equation and then proceed to prove global well-posedness (on the unit circle). Further well-posedness results are obtained for weak solutions of (1) among which is the Dirac delta function momentum (\(m=\delta\)) wave solution (that is analogous to the CH peakon wave). It is known that the CH equation has solutions which lead to blowup in finite time. Interestingly, this study shows that the solution dynamics for the mCH show significant differences to those for the CH equation. For example, it turns out that the mCH (1) does not admit finite-time blowup solutions for all \(k\geq2\). The study concludes with observations on the whole real-line case and some numerical simulations of (solitary-wave) solutions. This study will be of interest to those working on nonlinear wave equations -- especially shallow-water wave equations like the CH equation -- and the analytical and dynamical properties of their solutions.
scientific article; zbMATH DE number 5551136
scientific article