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Hayato Miyazaki

Publications and source records attributed to Hayato Miyazaki.

23 records · Page 2Linked to original sources
arXiv2017

Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical homogeneous nonlinearity

We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.

math.AP↗
arXiv2017

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

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. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in [10]. Moreover, we present a candidate of the second asymptotic profile to the solution.

math.AP↗
arXiv2017

Global behavior of solutions to generalized Gross-Pitaevskii equation

This paper is concerned with time global behavior of solutions to nonlinear Schrödinger equation with a non-vanishing condition at the spatial infinity. Under a non-vanishing condition, it would be expected that the behavior is determined by the shape of the nonlinear term around the non-vanishing state. To observe this phenomenon, we introduce a generalized version of the Gross-Pitaevskii equation, which is a typical equation involving a non-vanishing condition, by modifying the shape of nonlinearity around the non-vanishing state. It turns out that, if the nonlinearity decays fast as a solution approaches to the non-vanishing state, then the equation admits a global solution which scatters to the non-vanishing element for both time directions.

math.AP↗
arXiv2016

Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity

In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity which is of the long range critical order and is not necessarily a polynomial, in one and two space dimensions. As the nonlinearity is the critical order, the possible asymptotic behavior depends on the shape of the nonlinearity. The aim here is to give a sufficient condition on the nonlinearity to construct a modified wave operator. To deal with a non-polynomial nonlinearity, we decompose it into a resonant part and a non-resonant part via the Fourier series expansion. Our sufficient condition is then given in terms of the Fourier coefficients. In particular, we need to pay attention to the decay of the Fourier coefficients since the non-resonant part is an infinite sum in general.

math.AP↗