Time splitting method for nonlinear Schrodinger equation with rough initial data in L2

초록

We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schr & ouml;dinger equation with rough initial data in L-2, {i partial derivative(t)u + Delta u = lambda|u|(p )u, (x, t) is an element of R(d )x R+, u(x, 0) = phi(x), x is an element of R-d, where lambda is an element of {-1,1} and p > 0. While the Lie approximation ZL is known to converge to the solution u when the initial datum phi is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data phi is an element of L-2(R-d), we prove the L-2 convergence of the filtered Lie approximation Z (flt) to the solution u in the mass-subcritical range, 0 < p < 4/d. Furthermore, we provide a precise convergence result for radial initial data phi is an element of L-2(R-d).

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