Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions

Asymptotic behavior in time of solutions to complex-valued nonlinear Klein-Gordon equation in one space dimension

“…Here, the asymptotic behavior of the solution is given by the solution to the linear Dirac equation with a logarithmic phase correction as in Theorem 1.1. [10], we studied large time behavior of complex-valued solutions to the Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two space dimensions:…”

“…We constructed a solution to (1.4) which converges to prescribed final states, where the final state is given by the free solution with a logarithmic phase correction. Note that the logarithmic phase correction given by [10] has one more parameter which is characterized by the final data. It is an interesting open question whether all small global solutions to (1.4) behave like such an asymptotic profile.…”

“…It is known that Φ − = Φ + for the real-valued case (see [2]). Note that in [10] we constructed a solution to (1.1) which converge to "prescribed" final states in the sense of (1.3).…”

On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension

“…A difference between this behavior and that for (1.3) is that the phase correction term Ψ ± depends on both Φ + and Φ − , which reflects the presence of an interaction between two components of the system. See also [16][17][18] for the twoand three-dimensional gauge invariant critical equation of the form (1.6).…”

On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension

“…A difference between this behavior and that for (1.3) is that the phase correction term Ψ ± depends on both Φ + and Φ − , which reflects the presence of an interaction between two components of the system. See also [15][16][17] for the two-and three-dimensional gauge invariant critical equation of the form (1.6).…”