Singular Limits of Klein-Gordon-Schrödinger Equations to Schrödinger-Yukawa Equations

In this paper, we study analytically and numerically the singular limits of the nonlinear Klein–Gordon–Schrödinger (KGS) equations in Rd (d = 1, 2, 3) both with and without a damping term to the nonlinear Schrödinger–Yukawa (SY) equations. By using the two-scale matched asymptotic expansion, formal limits of the solution of the KGS equations to the solution of the SY equations are derived with an additional correction in the initial layer. Then for general initial data, weak and strong convergence results are established for the formal limits to provide rigorous mathematical justification for the matched asymptotic approximation by using the weak compactness argument and the (modulated) energy method, respectively. In addition, for well-prepared initial data, optimal quadratic and linear convergence rates are obtained for the KGS equations both with and without the damping term, respectively, and for ill-prepared initial data, the optimal linear convergence rate is obtained. Finally, numerical results for the KGS equations are presented to confirm the asymptotic and analytic results.