Abstract
In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of derivative type: iut + uxx = N(u, ū, ux, ūx), t ∈ R, x ∈ R; u(0, x) = u0(x), x ∈ R, (A) where N(u, ū, ux, ūx) = K1\u\2u + K2\u\2ux + K3u2ūx + K4\ux\2u + K5ūu2x + K6\ux\2ux, the functions Kj = Kj(\u\2), Kj(z) ∈ C∞([0,∞)). If the nonlinear terms N = ūu2x/1+|u|2 then equation (A) appears in the classical pseudospin magnet model [16]. Our purpose in this paper is to consider the case when the nonlinearity N depends both on ux and ūx. We prove that if the initial data u0 ∈ H3,∞ and the norms ||u0||3,l are sufficiently small for any l ∈ N, (when N depends on ūx), then for some time T > 0 there exists a unique solution u ∈ C∞([-T,T]\{0};C∞(R)) of the Cauchy problem (A). Here Hm,s = {φ ∈ L2; ||φ||m,s < ∞}, ||φ||m,s = ||(1 + x2)s/2(1 - ∂2x)m/2φ||L2, Hm,∞ = ∩s≥1Hm,s.
Original language | English |
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Pages (from-to) | 685-695 |
Number of pages | 11 |
Journal | Discrete and Continuous Dynamical Systems |
Volume | 5 |
Issue number | 3 |
DOIs | |
Publication status | Published - 1999 Jul |
Externally published | Yes |
Keywords
- Derivative type
- Nonlinear Schrödinger
- Smoothing effects
ASJC Scopus subject areas
- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics