We study the Cauchy problem for the nonlinear Schrödinger equations with dissipative nonlinearity. In particular, we show the time decay estimate for global solutions in Lebesgue space of square integrable functions. Main results of this paper are the extension of results that have been proved in the work of Hayashi et al. [Adv. Math. Phys. 5, 3702738 (2016)].

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We prove the global existence of analytic solutions to the Cauchy problem for a system of nonlinear Schrödinger equations with quadratic interaction in space dimension n 3 under the mass resonance condition. Lagrangian formulation is also described. (2010): 35Q55. Mathematics subject classification

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We consider the global Cauchy problem for a system of Schrödinger equations with quadratic interaction. Two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition. One is the usual analytic smoothing effect in space variables in terms of the generator of Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to Galilei generators for sufficiently small data with exponential decay at infinity in space ℝn with n ≥ 3. The other is analytic smoothing effect in space-time variables in terms of generator of pseudo-conformal and Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to pseudo-conformal and Galilei generators for sufficiently small data with exponential decay in ℝ4. We also discuss the associated Lagrange structure.