New Darboux Transformation for Hirota-Satsuma coupled KdV System

A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV system. It is shown that this Darboux transformation can be constructed by means of two methods: Painlev\'{e} analysis and reduction of a binary Darboux transformation. By iteration of the Darboux transformation, the Grammian type solutions are found for the coupled KdV system.Comment: LaTeX 2.09, 10 page

Numerical integration of coupled Korteweg-de Vries System

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes. The method is applied to Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to non-integrable cases.Comment: 11 pages, 13 figure