MODIFIED SCATTERING FOR THE QUADRATIC NONLINEAR KLEIN-GORDON EQUATION IN TWO DIMENSIONS.

In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein-Gordon equation (NLKG) in two space dimensions: ( + 1)u = λ|u|u, t ∈ R, x ∈ R2, where = ∂t² - Δ is d'Alembertian. For a given asymptotic profile uap, we construct a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap 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. [ABSTRACT FROM AUTHOR]