Asymptotic behavior in time of solution to system of cubic nonlinear Schr"odinger equations in one space dimension

Abstract

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Schrödinger equations in one space dimension. It turns out that for a system there exists a small solution of which asymptotic profile is a sum of two parts oscillating in a different way. This kind of behavior seems new. Further, several examples of systems which admit solution with several types of behavior such as modified scattering, nonlinear amplification, and nonlinear dissipation, are given. We also extend our previous classification result of nonlinear cubic systems.

Polynomial deceleration for a system of cubic nonlinear Schrödinger equations in one space dimension

“…This system is also studied in [18] and a similar slowly-decaying solution is found. However, an ODE analysis shows that the mechanism of the appearance of the logarithmic amplitude correction is slightly different.…”

“…A typical example is (1.1) with the choice λ 6 = λ 1 ( = 0) or λ 6 = 3λ 1 ( = 0). The second, third, and fourth authors show in [18] that (1.1) with these two choices admits a solution of which asymptotic profile is the pair of…”

“…As for (1.1), the case (λ 6 − λ 1 )(λ 6 − 3λ 1 ) = 0 is a threshold. Indeed, it is proved in [18] that the nonlinear amplification do not take place in the case (λ 6 − λ 1 )(λ 6 − 3λ 1 ) > 0. Further, it is revealed that the asymptotic behavior of u 2 in this case is a sum of two parts which oscillate in a different way.…”

“…The existence of blow-up solutions is studied also in [3,4,12] without a smallness assumption on the data. It is also known that there exists a system of the form (1.3) (with real coefficients) for which strong nonlinear amplification phenomenon of this type occurs (see [18]).…”