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.