اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNonlinear coherent states and Ehrenfest time for Schrodinger equation
346
0
0.0
(
0
)
تأليف
Remi Carles
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.
قيم البحث
اقرأ أيضاً
We consider the propagation of wave packets for a one-dimensional nonlinear Schrodinger equation with a matrix-valued potential, in the semi-classical limit. For an initial coherent state polarized along some eigenvector, we prove that the nonlinear
evolution preserves the separation of modes, in a scaling such that nonlinear effects are critical (the envelope equation is nonlinear). The proof relies on a fine geometric analysis of the role of spectral projectors, which is compatible with the treatment of nonlinearities. We also prove a nonlinear superposition principle for these adiabatic wave packets.
In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrodinger equations (NLS) that we gave in cite{CM15APDE}. We consider a NLS with a Schrodinger operator with
several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the refined profile, a quasi--periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.
The long-time asymptotic behavior of solutions to the focusing nonlinear Schrodinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the presence of discre
te spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the $xt$-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations.
In this article, we prove the scattering for the quintic defocusing nonlinear Schrodinger equation on cylinder $mathbb{R} times mathbb{T}$ in $H^1$. We establish an abstract linear profile decomposition in $L^2_x h^alpha$, $0 < alpha le 1$, motivated
by the linear profile decomposition of the mass-critical Schrodinger equation in $L^2(mathbb{R}^d )$, $dge 1$. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrodinger system, whose scattering can be proved by using the techniques in $1d$ mass critical NLS problem by B. Dodson, to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method. As a byproduct of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrodinger system, we also prove the conjecture of the global well-posedness and scattering of the two-discrete-component quintic resonant nonlinear Schrodinger system made by Z. Hani and B. Pausader [Comm. Pure Appl. Math. 67 (2014)].
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic regio
n of space time $|x| < 2t$ for large times and provide bounds for the error which decay as $t to infty$ for a general class of initial data whose difference from the non-vanishing background possesss a fixed number of finite moments and derivatives. Using properties of the scattering map for (GP) we derive as a corollary an asymptotic stability result for initial data which are sufficiently close to the N-dark soliton solutions of (GP).
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
