On the singular solutions of the Korteweg-de Vries equation (Q650341)

The paper addresses the problem of existence of regular solutions of the Korteweg-de Vries equation. The author proves two non-existence results. The first one states that there do not exist nonsingular solutions of the KdV equations in a rectangular domain \(|x|<L\), \(0\leq t\leq T\) satisfying the initial condition \(u(x,0)=u_0(x)\) and the inequality \(|uu_x|\leq q |u_x|^2\) with some \(q<1\), provided \(u_0(x)\) satisfies a certain integral estimate and \(T\) is large enough. The second results concerns the nonexistence of a bounded traveling wave solution whose parameter satisfies certain restrictions.