Solitary waves for the Hartree equation with a slowly varying potential

Abstract

We study the Hartree equation with a slowly varying smooth potential, V (x) = W (hx), and with an initial condition that is ε ≤ √ h away in H 1 from a soliton. We show that up to time |log h|/ h and errors of size ε + h 2 in H 1 , the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer and Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the result of Holmer and Zworski to more general initial conditions.

Benjamin–Ono soliton dynamics in a slowly varying potential

“…In [16], solutions are shown to remain h 2 close to a solitary wave profile in the energy space H 1 on the time scale δh −1 log h −1 . Datchev & Ventura [7] treated the case of the Hartree nonlinearity, and de Bouard & Fukuizumi [10] considered a stochastic perturbation of NLS. Fractional NLS equations have been considered by Secchi & Squassina [36], and a variational approach has been employed to study NLS in Benci, Ghimenti & Micheletti [2].…”

Phase-Driven Interaction of Widely Separated Nonlinear Schrödinger Solitons

“…Item (1) is immediate from the definition of Π (1,0,0,0) . For item (2), we recall that ker S = span{∂ a φ,∂ θ φ} and moreover, 0 is an isolated point in the spectrum of S. Thus S −1 : (ker S) ⊥ → (ker S) ⊥ is bounded as an operator on L 2 . The inequality (4.17) follows from this and elliptic regularity.…”

Stability of standing waves for theL2-critical Hartree equations with harmonic potential

“…As we all know, one of the most interesting issues concerning the Cauchy problem for this equation is the stability or instability of the standing waves. For N ¼ 3, 5 0 and ¼ 1, the solitary waves of Equation (1.1) have been discussed by Datchev and Ventura [7]. When 5 0 and 2 5 5 min{4, N}, Wang proved the strong instability of standing waves of Equation (1.1) under an appropriate assumption of the frequency.…”

“…Email: hjmath@163.com As we know, if 4 0 (repulsive interactions), Equation (1.1) describes the BoseEinstein condensation (BEC) in gases with very weak repulsive two-body interactions which can be found in systems of 87 Rb or 23 Na atoms [1,2]. If 5 0 (attractive interactions), BEC in gases with very weak attractive two-body is observed, for example, in systems of 7 Li atoms, as long as the gas in the trap has a sufficiently low density [1]. That is to say, if the number of condensed particles provided, say M ¼ R R N j'j 2 dx, is below a critical value M c , the BEC with attractive interactions is metastable.…”

“…In fact, this problem is one of the most interesting topics both in mathematics and physics (see e.g. [3][4][5][6][7][8][9][10][11][12]). Now let us review some results concerning the Equation (1.1) (see [13] and the references therein).…”