Hironobu Sasaki

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (∂t + α∂x + iβ)Φ = λ|Φ|p−1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(−t) and t∂x + x∂t − α/2, where {D(t)}t∈ℝ is the free Dirac evolution group. For the readers convenience, in an appendix we list and prove fundamental properties of D(t)xD(−t) and t∂x + x∂t − α/2.

Weak limit theorem for a nonlinear quantum walk

This paper continues the study of large time behavior of a nonlinear quantum walk begun in Maeda et al. (Discrete Contin Dyn Syst 38:3687–3703, 2018). In this paper, we provide a weak limit theorem for the distribution of the nonlinear quantum walk. The proof is based on the scattering theory of the nonlinear quantum walk, and the limit distribution is obtained in terms of its asymptotic state.

Scattering problems for the one-dimensional nonlinear Dirac equation with power nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (∂t + α∂x + iβ) = λ||p−1. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space.

Inverse Scattering for the Nonlinear Schrödinger Equation with the Yukawa Potential

We study the inverse scattering problem for the three dimensional nonlinear Schrödinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.

Remark on inverse scattering for the Schrödinger equation with cubic convolution nonlinearity

We discuss inverse scattering for the nonlinear Schrodinger equation. The nonlinearity of the equation is an approximate expression of the nonlocal nonlinearity. We give a reconstruction formula for determining the nonlinearity by the knowledge of given scattering states.

Scattering operator for the one-dimensional Dirac equation with power nonlinearity

We consider the nonlinear Dirac equation with a power nonlinearity (∂t + α∂x + iβ)Φ = λ|Φ|p−1Φ, where 3 < p < 5, λ ∈ ℂ. We prove the existence of the scattering operator in the neighborhood of the origin of the weighted Sobolev space H1,1 ∩ Hs,0, where 4/(p − 1) < s < p.