Complete integrability and singularity confinement of nonautonomous modified Korteweg-de Vries and Sine Gordon mappings

A systematic investigation on the complete integrability of the nonautonomous discrete–discrete modified Korteweg–de Vries (ΔΔmKdV) and sine Gordon (ΔΔsG) mappings is presented using Lax pair technique and singularity confinement criteria. We derive conditions on the coefficients of Lax matrices such that the ΔΔmKdV and ΔΔsG admit a Lax representation separately. We then demonstrate that the obtained conditions for each of the nonautonomous equations are indeed precisely the necessary conditions to satisfy an ultra-local version of singularity confinement criteria. A similarity reduction of both the ΔΔmKdV and ΔΔsG is obtained and shown to lead a new discrete Painleve type of equations of higher order possessing Lax representation.