## Abstract

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 iu_{t} + 1/2u_{xx} + script N = 0, (t, x) ∈ ℝ × ℝ; u(0, x) = u_{0}(x), x ∈ ℝ, (A) where script N = λ|u|^{2}u + 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)| ≤ Cz^{1+σ-j}, j = 0, 1, 2, 3, for z → +0, where σ > 0, and g ∈ C^{5} ([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 ∥u_{0}∥ _{H}3.0 + ∥u_{0}∥ _{H}0.3 is sufficiently small, where H^{m,s} = {φ ∈ S′; ∥φ∥_{m,s} = ∥(1+x^{2})^{s/2}(1 - ∂^{2}_{x})^{m/2}φ∥ _{L}2 < ∞}, 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 language | English |
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Pages (from-to) | 1415-1425 |

Number of pages | 11 |

Journal | Nonlinearity |

Volume | 12 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1999 Sept |