Asymptotic behavior in time of solutions to the derivative nonlinear Schrodinger equation

We study the asymptotic behavior in time of solutions to the Cauchy problem for the derivative nonlinear Schrodinger equation [GRAPHICS] where a is an element of R. We prove that if \\u(0)\\(H1.2) + \\u(0)\\(H3.0) is sufficiently small, then the solution of (DNLS) satisfies the time decay estimate \\u(t)\\(L) infinity less than or equal to C(1+\t\)(-1/2), where H-m,H-s = {f is an element of S'; \\f\\m,s = \\(1 + \x\(2))(s/2)(1-partial derivative(x)(2))(m/2)f\\(L)2 < infinity}, m, s is an element of R, The above L-infinity time decay estimate is very important for the proof of existence of the modified scattering states to (DNLS), In order to derive the desired estimate we introduce a certain phase function. (C) Elsevier, Paris.