Scattering problem and asymptotics for a relativistic nonlinear Schrodinger equation

We study the global existence and asymptotic behaviour in time of solutions of the Cauchy problem for the relativistic nonlinear Schrodinger equation in one space dimension iu(t) + 1/2 u(xx) + N = 0, (t, x) is an element of R x R; u(0, x) = u(0)(x), x is an element of R, (A) where N = lambda\u/(2)u + uf(\u\(2)) - ug'(g\u\(2))(g(\u\(2)))(xx). lambda is an element of R, the real-valued functions f and g are such that \f((j))(z)\ less than or equal to Cz(1+sigma-j), j = 0, 1,2, 3, for z --> +0, where sigma > 0, and g is an element of C-5([0, infinity)). Equation (A) models the self-channelling of a high-power, ultra-short laser in matter if f(z) = 2 lambda(1 - z/2 - (1 + z)(-1/2)), g(z) = root 1+z, for all z greater than or equal to 0. When la mbda = 0, f = 0 equation (A) also has some applications in condensed matter theory, plasma physics, Heisenberg ferromagnets and fluid mechanics. We pr ove that if the norm of the initial data parallel to u(o)parallel to(H)3,0parallel to u(o)parallel to(H)0,3 is sufficiently small, where H-m,H-s = {p hi is an element of S'; parallel to phi parallel to(m,s), = parallel to(1 x(2))(s/2) (1 - partial derivative(x)(2))(m/2)phi parallel to(L)2 < infini ty}, then the solution of the Cauchy problem (A) exists globally in time an d satisfies the sharp L-infinity time-decay estimate parallel to u(t)parall el to L infinity less than or equal to C(1 + \t\)(-1/2). Furthermore, we pr ove the existence of the modified scattering states and the nonexistence of the usual scattering states by introducing a certain phase function when l ambda not equal 0. On the other hand, the existence of the usual scattering states when lambda = 0 follows easily from our results.