scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

scholarly journals Forced cubic Schrödinger equation with Robin boundary data: continuous dependency result

2000 ◽  
Vol 41 (3) ◽  
pp. 301-311 ◽  
Author(s):  
Charles Bu
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Boundary Data ◽  
Schrodinger Equation ◽  
Cubic Schrödinger Equation ◽  
Robin Boundary

AbstractFor the cubic Schrödinger equation iut = uxx + k|u|2u, 0 ≤ x, t < ∞, initial data u(x, 0) = u0(x) ∈ H2[0, ∞), and Robin boundary data ux(0, t) + αu(0, t) = R(t) ∈ C2[0, ∞) (where α is real), we show that the solution u depends continuously on u0 and R.

Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide ℝ2 × 𝕋2

2019 ◽  
Vol 16 (01) ◽  
pp. 73-129 ◽  
Author(s):  
Zehua Zhao
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Data ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Resonant System ◽  
Induction Method ◽  
Profile Decomposition ◽  
Cubic Schrödinger Equation ◽  
Cubic Nonlinear Schrödinger Equation

We consider the problem of large data scattering for the defocusing cubic nonlinear Schrödinger equation on [Formula: see text]. This equation is critical both at the level of energy and mass. The key ingredients are global-in-time Stricharz estimate, resonant system approximation, profile decomposition and energy induction method. Assuming the large data scattering for the 2d cubic resonant system, we prove the large data scattering for this problem. This problem is the cubic analogue of a problem studied by Hani and Pausader.

scholarly journals Solitary waves for nonlinear Schrödinger equation with derivative

2018 ◽  
Vol 20 (04) ◽  
pp. 1750049 ◽  
Author(s):  
Changxing Miao ◽  
Xingdong Tang ◽  
Guixiang Xu
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Solitary Waves ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Critical Parameters ◽  
Well Posedness ◽  
Nonlinear Schrödinger ◽  
Global Result

In this paper, we characterize a family of solitary waves for nonlinear Schrödinger equation (NLS) with derivative (DNLS) by the structure analysis and the variational argument. Since DNLS does not enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters [Formula: see text] and the critical parameters [Formula: see text], we show the existence and uniqueness of the solitary waves for DNLS, up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters [Formula: see text], [Formula: see text] and the supercritical parameters [Formula: see text], there is no nontrivial solitary wave for DNLS. At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for DNLS with initial data in the invariant set [Formula: see text], with [Formula: see text], [Formula: see text] or [Formula: see text]. On the one hand, different with the scattering result for the [Formula: see text]-critical NLS in [B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285(5) (2015) 1589–1618], the scattering result of DNLS does not hold for initial data in [Formula: see text] because of the existence of infinity many small solitary/traveling waves in [Formula: see text] with [Formula: see text], [Formula: see text] or [Formula: see text]. On the other hand, our global result improves the global result in [Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. Partial Differential Equations 6(8) (2013) 1989–2002; Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. Partial Differential Equations 8(5) (2015) 1101–1112] (see Corollary 1.6).

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