High-order asymptotic expansion for the nonlinear Klein-Gordon equation in the non-relativistic limit regime

This paper presents an investigation into the high-order asymptotic expansion for 2D and 3D cubic nonlinear Klein-Gordon equations in the non-relativistic limit regime. There are extensive numerical and analytical results concerning that the solution of NLKG can be approximated by first-order modulated Schrödinger profiles in terms of $e^{i\frac{t}{{\epsilon }^2}}v+c.c.$, where v is the solution of related NLS and “$c.c.$” denotes the complex conjugate. Particularly, the best analytical result up to now is given in [20], which proves that the $L_x^2$ norm of the error can be controlled by ${\epsilon }^2+{\left({\epsilon }^{2}t\right)}^{\frac{\alpha }{4}}$ for $H_x^{\alpha }$-data, $\alpha \in [1,4]$. As for the high-order expansion, to our best knowledge, there are only numerical results, while the theoretical one is lacking.

In this paper, we extend this study further and give the first high-order analytical result. We introduce the high-order expansion inspired by the numerical experiments in [24], [15]:$e^{i\frac{t}{{\epsilon }^2}}v+{\epsilon }^2(\frac{1}{8}{e}^{3i\frac{t}{{\epsilon }^2}}{v}^3+e^{i\frac{t}{{\epsilon }^2}}w)+c.c.,$ where w is the solution to some specific Schrödinger-type equation. We show that the $L_x^2$ estimate of the error is of higher order ${\epsilon }^4+{\left({\epsilon }^{2}t\right)}^{\frac{\alpha }{4}}$ for $H_x^{\alpha }$-data, $\alpha \in [4,8]$. Besides, some counter-examples are given to suggest the sharpness of this upper bound.