The initial value problem for nonlinear Schrödinger equations with a dissipative nonlinearity in one space dimension

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

“…There have been some research about decay estimates of solutions to the subcritical nonlinear Schrödinger equation (1.1) with V (x) ≡ 0 and Imλ < 0 for arbitrarily large initial data (see e.g. [21] and [23]). 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].…”

The Effect of Gain and Strong Dissipative Structures on Nonlinear Schrödinger Equations in Optical Fiber

“…where ≥ 0, ∈ R, > 1, and ∈ C. The nonlinearity |V| −1 V with I < 0 and |I | > (( − 1)/2√ )|R | is called strong dissipative. In [3,4], the large initial problem for (3) with the strong dissipative nonlinearities was investigated. The global solution V( , ) to (3) decays like −1/2 (log ) −1/2 in the sense of ∞ for > 1, when = 3.…”

“…The global solution V( , ) to (3) decays like −1/2 (log ) −1/2 in the sense of ∞ for > 1, when = 3. Moreover ‖V‖ ∞ ≤ −1/( −1) for > 1, if 2.686 ≈ (5 + √ 33)/4 < < 3 in [3] and 2.586 ≈ (19 + √ 145)/12 < < 3 in [4], respectively. To study the time decays of solutions to (3), the estimate of ‖F (− )V‖ ∞ is useful.…”

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

“…x u = λ|u| q u (1.4) in one space dimension, where x ∈ R, t > 0, λ ∈ C \{0}, q = 2, was studied in [24] for small initial data in the case of Imλ < 0. If q < 2 and q is close to 2, (1.4) was investigated in [15] under the dissipative condition such that Imλ < 0 with the smallness conditions on the initial data, and in [16] and [12] under the strong dissipative conditions such that Imλ < 0 and |Imλ| ≥ q 2 √ q+1 |Reλ| without smallness conditions on the initial data. In these cases, nonlinear effects of the equation (1.4) are considered as dissipative ones.…”