Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation

In this paper the Cauchy problem for the Korteweg–de Vries equation $u_t+u_{xxx}=uu_x$, $x\in\mathbf R^1$, $0<t<T$, with initial condition $u(0,x)=u_0(x)$ is considered in nonlocal formulation. In the case of an arbitrary initial function $u_0(x)\in L^2(\mathbf R^1)$ the existence of a generalized $L^2$-solution is proved, and its smoothness is studied for $t>0$. A class of well-posed solutions is distinguished among the generalized solutions under consideration, and within this class theorems concerning existence, uniqueness and continuous dependence of solutions on initial conditions are proved.
Bibliography: 28 titles.