Integrability study and numerical analysis of a Korteweg-de Vries equation with variable coefficients (Q917809)

The authors discuss the Korteweg-de Vries equation (KdV) with time (t) dependent coefficients, \(u_ t+\alpha t^ nuu_ x+\beta t^ mu_{xxx}=0,\) where the subscripts denote partial derivatives, m and n are non-negative integers and \(\alpha\) and \(\beta\) are constants. They show that under a transformation of variables the cases \(n=m\) and \(m=2n+1\) are equivalent to the KdV and cylindrical KdV (cKdV) respectively. They also present numerical results for the cases \(m=n=1\); \(m=1\), \(n=0\); and \(m=2\), \(n=1\). Only the first example, equivalent to the KdV, exhibits the usual soliton properties (preservation of shape during translation and after interaction). In the other cases, even though the first is equivalent to the integrable cKdV, soliton-like profiles do not persist but develop oscillatory tails during translation and interaction, appearing to be qualitatively similar in behaviour.