We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrodinger equations of the following form where Jf(M, u, w x ,M x )=Jfi!w| 2 M-h/Jf 2 |w| 2 M x +zJf 3 w 2 w x + Jf4 u x \ 2 u-T^r 5 u Jf, = jr y (|«| 2 ), jT / (z)*EC 3 (R + ) ; JfXz)=A,+O(z), as z^ + 0, jf" JT 6 are real valued functions. Here the parameters Ai, A 6 £R, and A 2 , A 3 , A 4 , A 5 £C are such that A 2 -/U^R and A 4 -A 5 GR. If and A 5

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... l, ^1 = ^2 = ^3 = Jf 4 =Jf 6 = 0 equation (A) appears in the classical pseudospin magnet model [9] . We prove that if u 0^H *' 0l^H2 ' l and the norm llwolls, o+ llwolk i=e is sufficiently small, then the solution of (A) exists globally in tune and satisfies the sharp time decay estimate ||ii(Olko b -^Ce(l+ \t I)' 172 , where ML, s , p HI(l+x 2 ) s/2 (l-d^m /2 *= {<p£E:S' ; \\<P\\m, s ,p< c°} -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. § 1. Introduction In this paper we study the Cauchy problem for the derivative cubic nonlinear