Semi-Classical Wave Packet Dynamics for Hartree Equations

Abstract

We study the propagation of wave packets for nonlinear nonlocal Schrödinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.

Interaction of Coherent States for Hartree Equations

“…As established in [9], if ψ ε (0, ·) ∈ L 2 (R d ), then under Assumption 1, (1.1) has a unique solution ψ ε ∈ C(R + ; L 2 (R d )), regardless of the value of α, 1 and…”

“…This is in sharp contrast with the case of a homogeneous kernel, K (x) = λ|x| −γ , 0 < γ < min (2, d). It was shown in [9] that in this case, the critical value for α is α c = 1 + γ /2, and that when α = α c , the superposition principle remains, even though the nonlinearity affects the propagation of a single wave packet at leading order (the envelope equation is nonlinear).…”

“…Technically, the reason is two-fold. First, in [3,9], it is assumed that ∇ K (0) = 0, so the first line in (9.4) vanishes. Then, the second line in (9.4) accounts for the presence of two wave packets, and measures some coupling through a phase modulation; it vanishes in the case of a single wave packet.…”

“…Since both cases are rather similar, we describe the case of a Hartree nonlinearity only [9]. The main difference in the setting is that (1.1) is replaced with…”

“…We have three sets of terms, involving ∇u 1 , y 1 u 1 and u 1 , respectively, so they must be treated separately. As in [9], we then modify the general strategy, and allow b 0 j to depend on ε. We alter the hierarchy as follows:…”