A coupled system of mKdV-KdV equations with damping

In this paper, we study the influence of a damping term on the dynamics of a coupled Korteweg–de Vries type system, namely the mKdV–KdV system, posed on the real line. The presence of damping plays a significant role in the long-time behavior of solutions and requires refined analytical techniques. By employing suitable energy estimates in Sobolev spaces and analytic function spaces, we first establish the local well-posedness of the damped coupled mKdV–KdV system for analytic initial data. The analysis is carried out in a framework that allows us to control both regularity and analyticity of solutions. Furthermore, by combining the local existence theory with an analytic approximate conservation law adapted to the damped setting, we extend the local solutions globally in time. As an additional qualitative result, we prove that the radius of spatial analyticity of solutions does not degenerate over time, but instead remains uniformly bounded below by a positive constant for all times. This demonstrates the stabilizing effect of the damping mechanism on the analytic structure of the solutions.