Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity

Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions

“…We mention similar but slightly different expansion of a nonlinearity into a infinite Fourier sires is used by the first author and Miyazaki [21] in the context of nonlinear Schrödinger equation. Since both of the resonant and non-resonant parts are O(t −1 ) in L 2 x , we need to cancel out those terms by the linear part, otherwise (2.2) fails.…”

Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions

“…We suppose that the nonlinearity F is homogeneous of degree 1 + 2/d, that is, F satisfies (1.1) F (λu) = λ 1+ 2 d F (u) for any u ∈ C and λ > 0. This is the continuation of the previous study in [10]. In [10], we consider one-and two-dimensional cases and give a sufficient condition on F : C → C for existence of a modified wave operator, that is, for that (NLS) admits a nontrivial solution which asymptotically behaves like…”

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In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [10], the first and the second authors consider one-and two-dimensional cases and gave a sufficient condition on the nonlinearity for that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converges to a given asymptotic profile in a faster rate than in the lower dimensional cases.

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Long Range Scattering for Nonlinear Schrödinger Equations with Critical Homogeneous Nonlinearity

“…More precisely, the behavior of a solution depends on the shape of the nonlinearity [3,7,8,15,16,18]. In [11,12], we introduce a decomposition of the nonlinearity (1.2)…”

“…According to these facts, we do not try to give a behavior in terms of {g n } n in this paper, but instead deny the existence of a solution that behaves like a free solution or a free solution with a logarithmic phase correction, that is, behaves like (1.4). This is a complementary study of [11,12], and is an extension of [17,19].…”