Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values
The paper deals with the initial-value problem for the Korteweg-de Vries (KdV) equation \(\partial_t u+ u\partial_x u+ \partial^3_x u= 0\), \(u(x, 0)= \phi(x)\), \(x\in \mathbb{R}\), \(t\in \mathbb{R}\). There arises a natural nonlinear map \(K: \phi\to u\) from the space \(H^s(\mathbb{R})\) to the space \(M= C([- T, T]; H^s(\mathbb{R}))\). It is proved in the paper that the map \(K\) is infinitely many times Fréchet differentiable; moreover, it is shown that \(K\) has a Taylor series expansion at any given \(\phi\in H^s(\mathbb{R})\). In contrast, the corresponding map \(K_p\) for the periodic KdV equation with \(x\in S\), where \(S\) is the unit length circle in the plane, is known to be only continuous from \(H^s(\mathbb{R})\) to \(M\). Now, the author proves that \(K_p\) is Lipschitz continuous from \(H^{s+ 1}(S)\) to \(C([- T, T]; H^s(S))\) and \(n\) times Fréchet differentiable from \(H^{s+ n- 1}(S)\) to \(C([- T, T]; H^s(S))\) for any \(n\geq 1\).
scientific article; zbMATH DE number 764028