The initial value problem for the sequence of generalized Korteweg-de Vries equations
The initial value problem for the nonlinear equation \[ (1)\quad \frac{\partial u}{\partial t}=K_ m(u),\quad u\quad periodic\quad in\quad x, \] which is the m-th member of the Korteweg-de Vries hierarchy is considered. Let \(H_ n(S^ 1)\) denote the classical Sobolev space with the norm \(\| v\|^ 2_ n=\sum_{\ell =n}\int^{1}_{0}| D^{\ell}v(x)|^ 2 dx\). The main result is the following Theorem. For each m, if \(n\geq 2m-1\) and the initial datum g is an element of \(H_ n(S^ 1)\) then there exists a global solution u(x,t) to (1) from \(H_ n(S^ 1)\). Furthermore, for \(n\geq 3m-2\), u(x,t) is uniquely determined by g. A similar study for the KdV equation which corresponds to \(m=2\) has been carried previously by Temam, Sjöberg and Kato.
scientific article; zbMATH DE number 3929572
scientific article