An extended class of nonlinear groups and its applications to the generalized Korteweg-de Vries equations (Q699964)

The paper deals with the initial value problem for the generalized Korteweg-de Vries equation \[ u_t+(f(u))_x+ u_{xxx}=0,\quad t,x\in {\mathbb{R}}. \] Two operators \(A\) and \(B\) are introduced to represent the linear and the nonlinear differential operators above, and the problem reduces to a semilinear evolution problem \[ u'(t)=(A+B)u(t),\quad t>0;\quad u(0)=v \] in the Sobolev space \(H^2({\mathbb{R}}).\) The solution operators of this problem are obtained by applying a generation theorem for groups of locally Lipschitzian operators.