On the One-dimensional Cubic Nonlinear Schrödinger Equation below L2 Tadahiro Oh and Catherine Sulem
نویسنده
چکیده
In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic Nonlinear Schrödinger equation (NLS) on the real line R and on the circle T for solutions below the L-threshold. We point out common results for NLS on R and the so-called Wick ordered NLS (WNLS) on T, suggesting that WNLS may be an appropriate model for the study of solutions below L(T). In particular, in contrast with a recent result of Molinet [33] who proved that the solution map for for the periodic cubic NLS equation is not weakly continuous from L(T) to the space of distributions, we show that this is not the case for WNLS.
منابع مشابه
Numerical Simulation of Resonant Tunneling of Fast Solitons for the Nonlinear Schrödinger Equation
In a recent paper [4], we showed that the phenomenon of resonant tunneling, well known in linear scattering theory, takes place for fast solitons of the Nonlinear Schrödinger (NLS) equation in the presence of certain large potentials. Here, we illustrate numerically this situation for the one dimensional cubic NLS equation with two classes of potentials, namely the ‘box’ potential and a repulsi...
متن کاملAlmost Surely Local Well-posedness of the Cubic Nonlinear Schrödinger Equation below L²(t)
We consider the local well-posedness of the periodic cubic nonlinear Schrödinger equation with initial data below L(T). In particular, we prove that it is locally well-posed almost surely for the initial data in the support of the Gaussian measures on H(T) for each s > − 1 8 .
متن کاملOn the Blowup for the L-critical Focusing Nonlinear Schrödinger Equation in Higher Dimensions below the Energy Class
We consider the focusing mass-critical nonlinear Schrödinger equation and prove that blowup solutions to this equation with initial data in H(R), s > s0(d) and d ≥ 3, concentrate at least the mass of the ground state at the blowup time. This extends recent work by J. Colliander, S. Raynor, C. Sulem, and J. D. Wright, [13], T. Hmidi and S. Keraani, [21], and N. Tzirakis, [36], on the blowup of t...
متن کاملScattering for the Non-radial 3d Cubic Nonlinear Schrödinger Equation
Scattering of radial H solutions to the 3D focusing cubic nonlinear Schrödinger equation below a mass-energy threshold M [u]E[u] < M [Q]E[Q] and satisfying an initial mass-gradient bound ‖u0‖L2‖∇u0‖L2 < ‖Q‖L2‖∇Q‖L2 , where Q is the ground state, was established in Holmer-Roudenko [8]. In this note, we extend the result in [8] to non-radial H data. For this, we prove a non-radial profile decompo...
متن کاملResonant Hamiltonian Systems Associated to the One-dimensional Nonlinear Schrödinger Equation with Harmonic Trapping
We study two resonant Hamiltonian systems on the phase space L2(R → C): the quintic one-dimensional continuous resonant equation, and a cubic resonant system that appears as the modified scattering limit of a specific NLS equation. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scale...
متن کامل