Rational Solutions to the KdV Equation in Terms of Particular Polynomials

Abstract

Here, we construct rational solutions to the KdV equation by particular polynomials. We get the solutions in terms of determinants of the order $n$ for any positive integer $n,$ and we call these solutions, solutions of the order $n.$ Therefore, we obtain a very efficient method to get rational solutions to the KdV equation, and we can construct explicit solutions very easily. In the following, we present some solutions until order 10.