We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrödinger equations of the following form where are real valued functions. Here the parameters, and λ2, λ3, λ4, λ5 ∈ C are such that λ2-λ3 ∈ R and λ4-λ5 ∈ R. If and λ5=β=±1, equation (A) appears in the classical pseudospin magnet model . We prove that if and the norm ‖u0‖3,0+‖u0‖2,1=∊ is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate ‖u(t)‖2,0∞≤C∊(1+ |t|)-1/2, where. Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.
|Number of pages||13|
|Journal||Publications of the Research Institute for Mathematical Sciences|
|Publication status||Published - 1999 Jan 1|
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