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We discuss local existence and gain of regularity for semilinear Schrödinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schrödinger-type equations. In particular, the sharp Gårding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. (1991): 35Q55, 35B65, 35G25, 35S05 Mathematics Subject Classification

We present local and global existence theorems for cubic semilinear Schrodinger equations. Our new results are the improvement of our previous ones ([2] , [3] , [4]). The idea of the proof consists of the energy and the decay estimates. These equations do not allow the classical energy estimates. To avoid this difficulty, we make strong use of S. Doi's method for linear Schrodinger type equations. Combining cubic nonlinearity and S. Doi's method, we obtain the improved results.

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow.