On the Nonlinear Group Associated with a Nonlinear Dispersive Equation

摘要

An initial periodic boundary value problem for a nonlinear dispersive equation, which is a modified version of the Kortweg-de Vries equation, is studied from the point of view of the theory of semigroups of nonlinear operators. The initial boundary value problem is reduced to the abstract Cauchy problem for a nonlinear evolution equation in a function space, and it is shown that the evolution equation admits regular solutions which form a nonlinear semigroup consisting of differentiable operators on the function space. Furthermore, it is shown that the semigroup is embedded in a nonlinear group of C(1) diffeomorphisms on the function space and the group possesses various interesting properties.