The asymptotic behavior of solutions of the Korteweg-de Vries equation evolving from very irregular data
The behavior of solutions of Korteweg-de Vries equation \(u_ t-6uu_ x+u_{xxx}=0\) for initial conditions \(u(x,0)=\mu (x)\) being a measure is studied. It is assumed that \(\mu\) is such that \(\int^{\infty}_{- \infty}(1+| x|)^ N d| \mu | (x)<\infty\) for \(N=3\) or \(N=4\). The main result states that if n is the number of poles of the transmission coefficient of the Schrödinger equation \(-y''+\mu (x)y=k^ 2y\), then one can specify an n-soliton solution \(u_ s(x,t)\) such that for any \(x_ 0\), \[ \sup \{| u(x,t)-u_ s(x,t)|: x\geq x_ 0\}=O(t^{-1/3}). \] The case \(n=0\) is also under discussion.
scientific article; zbMATH DE number 4046284
scientific article