First integral blow-up solutions to complex-valued KdV equations

Abstract

We apply Lie theory to determine the aone-parameter point transformations which leave complex-valued Korteweg-de Vries equations invariant. The conserved vectors of the systems are constructed. We provide travelling wave reductions that lead to third-order ordinary differential equations. These equations are highly nonlinear to solve directly and, therefore, we establish their first integrals. The latter is of second-order and facilitates the analysis of the complex-valued system, as well as some interesting blow-up solutions.