On Schrödinger systems with cubic dissipative nonlinearities of derivative type

Abstract

Consider the initial value problem for systems of cubic derivative nonlinear Schrödinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under which the small data solution gains an additional logarithmic decay as t → +∞ compared with the corresponding free evolution.

Small data global existence for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions

“…Remember that 1D cubic case is another critical situation, that is, 3 = (1 + 2/d)| d=1 . However, we need several modifications to prove Theorem 2.1 because the approach of [26] relies heavily on one-dimensional nature. Another remark concerning this point is that the condition (b) above can be replaced by the following apparently weaker one:…”

“…In the 1D cubic case, only one action of J m is enough for getting desired a priori L 2 -bounds. This is the point where the one-dimensional nature (such as the imbedding [26]. However, since we are considering the problem in R 2 now, we have to use J m several times.…”

On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D

“…Segawa, Sunagawa and Yasuda considered a sharp lower bound for the lifespan of small solutions to the subcritical Schrödinger equation (1.1) with V (x) ≡ 0 and Imλ > 0 in the space dimension n = 1, 2, 3 in [30]. For the systems of nonlinear Schrödinger equations, the existence of modified wave operators to a quadratic system in R 2 was studied in [16], and initial value problem for a cubic derivative system in R was investigated in [24].…”

Long-Long or Long-Short Range Interactions of Nonlinear Schrödinger Systems in One Space Dimension

“…For the Schrödinger systems with long range interactions, asymptotic behavior of solutions was known on the final state problem (see, e.g., [21], [9]). There are some results on the initial value problem for the Schrödinger systems (see e.g., [8], [13], [14] and [18]) and time decay estimates of the solutions were studied in some critical cases. The existence of ground states for some nonlinear Schrödinger systems was investigated in [11].…”