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 journalArticlepeer-review

15 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 Sept


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