Smoothing effects for some derivative nonlinear schrödinger equations

Nakao Hayashi, Pavel I. Naumkin, Patrick Nicolas Pipolo

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6 Citations (Scopus)


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 languageEnglish
Pages (from-to)685-695
Number of pages11
JournalDiscrete and Continuous Dynamical Systems
Issue number3
Publication statusPublished - 1999 Jul


  • Derivative type
  • Nonlinear Schrödinger
  • Smoothing effects


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