||Satoshi Masaki, Jason Murphy, Jun-ichi Segata, Modified scattering for the 1d cubic NLS with a repulsive delta potential, International Mathematics Research Notices, in press, We consider the initial-value problem for the one dimensional cubic nonlinear Schr"odinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev space lead to global solutions that exhibit modified scattering..
||Jean-Claude Saut, Jun-ichi Segata, Asymptotic behavior in time of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion, Discrete and Continuous Dynamical Systems. Series A, 10.3934/dcds.2019009, 39, 1, 219-239, 2019.04, We consider the asymptotic behavior in time of solutions to the nonlinear Schr"odinger equation with fourth order anisotropic dispersion which describes the propagation of ultrashort laser pulses in a medium with anomalous time dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to this equation which scatters to a solution of the linearized equation..
||Satoshi Masaki, Jun-ichi Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions, Transactions of the American Mathematical Society, 10.1090/tran/7262, 370, 11, 8155-8170, 2018.04, In this paper, we consider the long time behavior of solution to the quadratic gauge invariant nonlinear Klein-Gordon equation in two space dimensions. For a given asymptotic profile, we construct a solution to this equation which converges to given asymptotic profile as t goes infinity. Here the asymptotic profile is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity..
||Satoshi Masaki, Jun-ichi Segata, Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application, SIAM Journal on Mathematical Analysis, 10.1137/17M1153893, 50, 3, 2839-2866, 2018.04, In this paper, we give an improvement of non-diagonal Strichartz estimates for Airy equation by using a Morrey type space.
As its applications, we prove the small data scattering and the existence of special non-scattering solutions, which are minimal in a suitable sense, to the mass-subcritical generalized Korteweg-de Vries equation. Especially, a use of the refined non-diagonal estimate removes several technical restrictions on the previous work about the existence of the special
||Satoshi Masaki, Jun-ichi Segata, Existence of a minimal non-scattering solution to the mass sub-critical generalized Korteweg-de Vries equation, Annales de l'Institut Henri Poincar'e/Analyse non lin'eaire, 10.1016/j.anihpc.2017.04.003, 35, 2, 283-326, 2018.04, In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical
generalized Korteweg-de Vries equation in the scale critical Fourier Lebesgue space. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to Fourier Lebesgue space-framework and approximation of solutions to the generalized KdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schr"odinger equation..
||Yung-Fu Fang, Chi-Kun Lin, Jun-ichi Segata, The fourth order nonlinear Schr"odinger limit for quantum Zakharov system, Zeitschrift f"ur angewandte Mathematik und Physik, ZAMP - Journal of Applied Mathematics and Physics, 10.1007/s00033-016-0740-1, 67, 6, article number 145, 2016.04, This paper is concerned with the quantum Zakharov system. We prove that when the ionic speed of sound goes to infinity, the solution to the fourth order Schr"odinger part of the quantum Zakharov system converges to the solution to quantum modified nonlinear Schr"odinger eqaution..
||Satoshi Masaki, Jun-ichi Segata, On well-posedness of generalized Korteweg-de Vries equation in scale critical hat Lr space, Analysis & PDE, 10.2140/apde.2016.9.699, 9, 3, 699-725, 2016.04, The purpose of this paper is to study local and global well-posedness of initial value problem for the generalized Korteweg-de Vries equation in Fourier Lebesgue space. We show (large data) local well-posedness, small data global well-posedness, and small data scattering for generalized KdV equation in the scale critical Fourier Lebesgue space. A key ingredient is a Stein-Tomas type inequality for the Airy equation, which generalizes usual Strichartz' estimates for Fourier Lebesgue space-framework..
||Jun-ichi Segata, Final state problem for the cubic nonlinear Schr"odinger equation with repulsive delta potential, Communications in Partial Differential Equations, 10.1080/03605302.2014.930753, 40, 2, 309-328, 2015.04, We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schr"odinger equation with repulsive delta potential. We shall prove that for a given small asymptotic profile, there exists a solution to this equation which converges to given asymptotic profile as t to infinity. To show this result we exploit the distorted Fourier transform associated to the Schr"odinger equation with delta potential..
||Jun-ichi Segata, Refined energy inequality with application to well-posedness for the fourth order nonlinear Schr"odinger type equation on torus, Journal of Differential Equations, 10.1016/j.jde.2012.02.016, 252, 11, 5994-6011, 2012.04, We consider the time local and global well-posedness for the fourth order nonlinear Schr"odinger type equation on the torus. The nonlinear term contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to this equation..
||Masaya Maeda, Jun-ichi Segata, Existence and Stability of standing waves of fourth order nonlinear Schr"odinger type equation related to vortex filament, Funkcialaj Ekvacioj, 10.1619/fesi.54.1, 54, 1, 1-14, 2011.04, In this paper, we study the fourth order nonlinear Schr"odinger type equation which is a generalization of the Fukumoto-Moffatt model that arises in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave is orbitally stable in Sobolev spaces..
||Jun-ichi Segata, On asymptotic behavior of solutions to Korteweg-de type equations related to vortex filament with axial flow, Journal of Differential Equations, 10.1016/j.jde.2008.03.031, 245, 2, 281-306, 2008.04, We study the global existence and asymptotic behavior in time of solutions to the Korteweg-de Vries type equation called as ``Hirota" equation. This equation is a mixture of cubic nonlinear Schr"odinger equation and modified Korteweg-de Vries equation. We show the unique existence of the solution for this equation which tends to the given modified free profile by using the two asymptotic formulae for some oscillatory integrals..
||Jun-ichi Segata, Modified wave operators for the fourth-order non-linear Schr"odinger-type equation with cubic non-linearity, Mathematical Methods in the Applied Sciences, 10.1002/mma.751, 29, 15, 1785-1800, 2006.04, We consider the scattering problem for the fourth order nonlinear Schr"odinger type equation. We show the existence of the modified wave operators for this equation by imposing the mean zero condition for the final data..
||Naoyasu Kita, Jun-ichi Segata, Time local well-posedness for the Benjamin-Ono equation with large initial data, Publications of the Research Institute for Mathematical Sciences, Kyoto University, 10.2977/prims/1166642062, 42, 1, 143-171, 2006.01, This paper studies the time local well-posedness of the solution to the Benjamin-Ono equation. Our aim is to remove smallness condition on the initial data which was imposed in Kenig-Ponce-Vega's work..
||Jun-ichi Segata, Remark on well-posedness for the fourth order nonlinear Schr"odinger type equation, Proceedings of the American Mathematical Society, 10.1090/S0002-9939-04-07620-8, 132, 12, 3559-3568, 2004.12, We consider the initial value problem for the fourth order nonlinear Schr"odinger type equation related to the theory of vortex filament. In this paper we prove the time local well-posedness for this equation in the Sobolev space..