Long-time Asymptotics of the One-dimensional Damped Nonlinear Klein–Gordon Equation

Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equations

“…Note that this is also a necessary condition for lower p and N, and then the above result is equivalent to uniform bound in H of global solutions (cf. [3,5]). For general (subcritical) power, however, it seems to be an open question both with the damping and without it.…”

“…In the one dimensional case N = 1, Côte, Martel and Yuan [5] proved that the above three types exhaust all solutions in the energy space. To the best of our knowledge, this is the first complete proof of the soliton resolution conjecture for a non-integrable equation without any size or symmetry restriction and with moving solitons, provided that the damping is acceptable in the conjecture.…”

“…The center points {z k } of solitons may depend on time, but not necessarily be defined for all t → ∞. Note that in [5] those solutions satisfying (1.12) are called K-solitons. This is because the more focus is put in this paper on the intermediate and transitive dynamics than the asymptotic behavior.…”

“…This property is lost as soon as the number of solitons becomes 3 or more. For example, [5] constructed the asymptotic 3-solitons in the form on R (N = 1)…”

“…If only the middle soliton is collapsed by the instability and disappears (decays), then after that the solution contains only the two solitons with the same signs, which attract each other. By using the soliton resolution of [5], as well as stability of decaying and blowup, it is easy to observe that there are solutions in which the middle soliton collapses first, and afterward the other two solitons together, but eventually a soliton in the middle appears as the final state. This behavior is essentially different from the 1soliton case (2) in Theorem 1.1, because the final soliton is not a remaining one of the three starting ones, but created by merger of the two of them.…”