Approximation of a quasilinear Schrödinger equation by a Klein–Gordon equation in space dimension 2

In this article, we study the nonlinear plasma wave equation $\[-\varepsilon^{2}\dfrac{\curpartial^{2}u_{\varepsilon}}{\curpartial t^{2}}+2\mathrm{i}\dfrac{\curpartial u_{\varepsilon}}{\curpartial t}+\Delta u_{\varepsilon}=\big(\dfrac{1}{\sqrt{1+|u_{\varepsilon}|^{2}}}-1\big)u_{\varepsilon}+\dfrac{\Delta(\sqrt{1+|u_{\varepsilon}|^{2}})}{\sqrt{1+|u_{\varepsilon}|^{2}}}u_{\varepsilon}\]$ with initial data $u_{\varepsilon}(\cdot,0)=u_{0}^{\varepsilon}(\cdot)\in H^{8}(\mathbb{R}^{2}),\ \curpartial_{t}u_{\varepsilon}(\cdot,0)=u_{1}^{\varepsilon}(\cdot)\in H^{7}(\mathbb{R}^{2})$ . We show that the Cauchy problem is locally well‐posed on an interval [0,T] where the time T is independent of ε if u₁^ε is small enough. Then, we demonstrate the strong convergence of u_ε towards the solution u of a nonlinear relativistic Schrödinger equation as ε goes to 0.