Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions

Abstract

In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: $(\Box +1)u=\lambda |u|u$, $t\in \mathbb {R}$, $x\in \mathbb {R}^{2}$, where $\Box =\partial _{t}^{2}-\Delta$ is d’Alembertian. For a given asymptotic profile $u_{\mathrm {ap}}$, we construct a solution $u$ to (NLKG) which converges to $u_{\mathrm {ap}}$ as $t\to \infty$. Here the asymptotic profile $u_{\mathrm {ap}}$ is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.