On global small amplitude solutions to systems of cubic nonlinear Klein–Gordon equations with different mass terms in one space dimension

The initial value problem for the cubic nonlinear Klein–Gordon equation

“…In paper [13], the nonlinearities were classified into two types, one of them can be treated by the method of normal forms [24] and the other reveals an additional time decay rate via the operator x∂ t +t∂ x [14]. This method was extended to a system of nonlinear Klein-Gordon equations by [26]. Some sufficient conditions on cubic nonlinearities were given in [3], which allow us to prove global existence and to find sharp asymptotics of small solutions to the Cauchy problem (1.1) with small and regular initial data having a compact support.…”

Global existence of small solutions for quadratic quasilinear Klein–Gordon systems in two space dimensions

“…Later on, in [10], Ozawa, Tsutaya and Tsutsumi announced that their proof can be extended to the quasilinear case and studied scattering of solutions. More recently, Sunagawa [13] studied systems of Klein-Gordon equations with possibly different masses, in two space dimensions, for quadratic semi-linear nonlinearities. Under a nonresonance assumption on the masses, he obtained global existence for small data.…”

“…More precisely, they show that adding to the unknown v a conveniently chosen nonlocal quadratic expression in v; they can eliminate the worst contributions to the right-hand side, thus reducing themselves to a short-range problem. The method of Tsutsumi [14] and of Sunagawa [13] relies on the same ideas, except that one eliminates only part of the long-range term, using local expressions of the unknowns. But the remaining quantities in the right-hand side are then ''null forms'', that enjoy better decay estimates than general quadratic forms, and allow one to obtain energy estimates as in the short-range case.…”

“…More recently, Sunagawa [13] studied systems of one dimensional Klein-Gordon equations, with semi-linear cubic nonlinearities, under a nonresonance assumption. We refer to these papers for more complete bibliographical references to the one dimensional problem.…”

Scattering Problem for the Supercritical Nonlinear Schrödinger Equation in 1d

“…Now these methods are very important tools in the field of nonlinear dispersive equations. The algebraic normal form method by [14], [19] was used for the study of the quadratic nonlinear Klein-Gordon equations in two space dimensions. Also the method of [15] was applied to quadratic nonlinear Schrö dinger equations in two space dimensions for the nonlinearity u 2 , see [3].…”