Scattering problem and asymptotics for a relativistic nonlinear Schrödinger equation

Anne De Bouard, Nakao Hayashi, Pavel I. Naumkin, Jean Claude Saut

Research output: Contribution to journalArticle

8 Citations (Scopus)


We study the global existence and asymptotic behaviour in time of solutions of the Cauchy problem for the relativistic nonlinear Schrödinger equation in one space dimension iut + 1/2uxx + script N = 0, (t, x) ∈ ℝ × ℝ; u(0, x) = u0(x), x ∈ ℝ, (A) where script N = λ|u|2u + uf (|u|2) - ug′(|u|2)(g(|u|2))xx, λ ∈ ℝ, the real-valued functions f and g are such that |f(j)(z)| ≤ Cz1+σ-j, j = 0, 1, 2, 3, for z → +0, where σ > 0, and g ∈ C5 ([0, ∞)). Equation (A) models the self-channelling of a high-power, ultra-short laser in matter if f(z) = 2λ(1 - z/2 - (1 + z)-1/2, g(z) = √1 + z, for all z ≥ 0. When λ = 0, f = 0 equation (A) also has some applications in condensed matter theory, plasma physics, Heisenberg ferromagnets and fluid mechanics. We prove that if the norm of the initial data ∥u0H3.0 + ∥u0H0.3 is sufficiently small, where Hm,s = {φ ∈ S′; ∥φ∥m,s = ∥(1+x2)s/2(1 - ∂2x)m/2φ∥ L2 < ∞}, then the solution of the Cauchy problem (A) exists globally in time and satisfies the sharp L time-decay estimate ∥u(t)∥ L∞ ≤ C (1 + |t|)-1/2. Furthermore, we prove the existence of the modified scattering states and the nonexistence of the usual scattering states by introducing a certain phase function when λ ≠ 0. On the other hand, the existence of the usual scattering states when λ. = 0 follows easily from our results.

Original languageEnglish
Pages (from-to)1415-1425
Number of pages11
Issue number5
Publication statusPublished - 1999 Sep 1
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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