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Last Updated :2024/07/25

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Master

Affiliation (Master)

  • Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

  • Faculty of Science　Mathematics　Mathematics

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Profile and Settings

Affiliation

  • Hokkaido University, Faculty of Science, Professor

Degree

  • Doctor of Science（2009/03 Kyoto University）

Profile and Settings

  • Name (Japanese)

    Masaki

  • Name (Kana)

    Satoshi

  • Name

    202101020882147147

Affiliation

  • Hokkaido University, Faculty of Science, Professor

Achievement

Research Interests

  • nonlinear Klein-Gordon equaiton   tine-global dynamics   nonlinear scattering problem   stability of standing wave   soliton   large-time behavior of solutions   nonlinear Schrodinger equation   dispersive equations   

Research Areas

  • Natural sciences / Mathematical analysis

Research Experience

  • 2023/04 - Today Hokkaido University Faculty of Science Professor
  • 2016/04 - 2023/03 Osaka University Graduate School of Engineering Science Associate Professor
  • 2012/10 - 2016/03 Hiroshima University Graduate School of Engineering Associate Professor
  • 2010/04 - 2012/09 Gakushuin University Faculty of Science Assistant Professor
  • 2009/04 - 2010/03 Tohoku University Graduate School of Information Sciences JSPS fellow

Education

  • 2006/04 - 2009/03  Kyoto University  Graduate School of Science
  • 2004/04 - 2006/03  Kyoto University  Graduate School of Science
  • 2000/04 - 2004/03  Kyoto University  Faculty of Science  Faculty of Science

Awards

  • 2017/12 Division of Functional Equations, Mathematical Society of Japan Hukuhara prize
     Harmonic analysis and variational method for time global analysis of solutions to non-linear dispersive equations

Published Papers

  • Naoyasu Kita, Satoshi Masaki, Jun-ichi Segata, Kota Uriya
    Nonlinear Analysis 230 113216 - 113216 0362-546X 2023/05
  • Satoshi Masaki, Hayato Miyazaki
    published online in Differential Equations and Dynamical Systems 2022/08 [Refereed]
     
    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.
  • Satoshi Masaki, Jun-ichi Segata, Kota Uriya
    Transactions of the American Mathematical Society, Series B 9 (18) 517 - 563 2022/06/08 [Refereed]
     

    In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order.

  • Masaru Hamano, Satoshi Masaki
    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 41 (3) 1415 - 1447 1078-0947 2021/03 [Refereed]
     
    In this paper, we consider the quadratic nonlinear Schrodinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrodinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.
  • Satoshi Masaki, Jun-ichi Segata, Kota Uriya
    JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 139 177 - 203 0021-7824 2020/07 [Refereed]
     
    In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed. (C) 2020 Elsevier Masson SAS. All rights reserved.
  • A survey on long range scattering for Schrödinger equation and Klein-Gordon equation with critical nonlinearity of non-polynomial type.
    Satoshi Masaki
    RIMS Kôkyûroku Bessatsu B82 103 - 135 2020/06 [Refereed]
  • Satoshi Masaki, Koki Sugiyama
    DIFFERENTIAL AND INTEGRAL EQUATIONS 33 (5-6) 247 - 256 0893-4983 2020/05 [Refereed]
     
    We consider the decay rate of solutions to nonlinear Klein-Gordon systems with a critical type nonlinearity. We will specify the optimal decay rate for a specific class of Klein-Gordon systems containing the dissipative nonlinearities. It will turn out that the decay rate which is previously found in some models is optimal.
  • Satoshi Masaki, Jason Murphy, Jun-ichi Segata
    ANALYSIS & PDE 13 (4) 1099 - 1128 1948-206X 2020 [Refereed]
     
    We consider the initial-value problem for the one-dimensional nonlinear Schrodinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes into a small solitary wave plus a radiation term that decays and scatters as t -> infinity. In particular, we establish the asymptotic stability of the family of small solitary waves.
  • Satoshi Masaki, Jason Murphy, Jun-Ichi Segata
    INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2019 (24) 7577 - 7603 1073-7928 2019/12 [Refereed]
     
    We consider the initial-value problem for the one-dimensional cubic nonlinear Schrodinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev space lead to global solutions that decay in L-infinity and exhibit modified scattering.
  • Satoshi Masaki, Hayato Miyazaki, Kota Uriya
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 371 (11) 7925 - 7947 0002-9947 2019/06 [Refereed]
     
    In this paper, we consider the final state problem for the nonlinear Schrodinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [SIAM J. Math. Anal. 50 (2018), pp. 3251-3270], the first and second authors consider one-and two-dimensional cases and give a sufficient condition on the nonlinearity so 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 converge 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 the aforementioned article. Moreover, we present a candidate for the second asymptotic profile of the solution.
  • Satoshi Masaki, Hayato Miyazaki
    DIFFERENTIAL AND INTEGRAL EQUATIONS 32 (3-4) 121 - 138 0893-4983 2019/03 [Refereed]
     
    We consider large time behavior of solutions to the nonlinear Schrodinger 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 vertical bar u vertical bar(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.
  • Rowan Killip, Satoshi Masaki, Jason Murphy, Monica Visan
    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 39 (1) 553 - 583 1078-0947 2019/01 [Refereed]
     
    We consider the mass-subcritical NLS in dimensions d >= 3 with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.
  • Satoshi Masaki
    ASYMPTOTIC ANALYSIS FOR NONLINEAR DISPERSIVE AND WAVE EQUATIONS 81 259 - 309 2019 [Refereed]
     
    In this article, we consider mass-subcritical Hartree equation. Scattering problem is treated in the framework of weighted spaces. We first establish basic properties such as local-wellposedness and criteria for finite-time blowup and scattering. Then, the first result is that uniform in time bound in critical weighted norm implies scattering. The proof is based on the concentration compactness/rigidity argument initiated by Kenig and Merle. By using the argument, existence of a threshold solution between small scattering solutions and other solutions is also deduced for the focusing model, which is the second result. The threshold is neither ground state nor any other standing wave solutions, as is known for the power type NLS equation.
  • Satoshi Masaki, Jun-ichi Segata
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 370 (11) 8155 - 8170 0002-9947 2018/11 [Refereed]
     
    In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein-Gordon equation (NLKG) in two space dimensions: (square + 1) u =lambda vertical bar u vertical bar u, t is an element of R, x is an element of R-2, where square = partial derivative(2)(t) - Delta is d'Alembertian. For a given asymptotic profile u(ap), we construct a solution u to (NLKG) which converges to u(ap) as t -> infinity. Here the asymptotic profile u(ap) 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
    COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 17 (4) 1595 - 1611 1534-0392 2018/07 [Refereed]
     
    In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: (square + 1)u = lambda vertical bar u vertical bar(2/3)u, t is an element of R, x is an element of R-3, where square = partial derivative(2)(t) - Delta is d'Alembertian. We prove that for a given asymptotic profile u(ap), there exists a solution u to (NLKG) which converges to u(ap) as t -> infinity. Here the asymptotic profile u(ap) 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 the combination of Fourier series expansion for the nonlinearity used in our previous paper [23] and smooth modification of phase correction by Ginibre and Ozawa [6].
  • Satoshi Masaki, Jun-ichi Segata
    ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 35 (2) 283 - 326 0294-1449 2018/03 [Refereed]
     
    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 (gKdV) equation in the scale critical (L) over cap (r) space where (L) over cap (r) = {f is an element of S'(R)vertical bar parallel to f parallel to ((L) over cap)r = parallel to (f) over cap parallel to(L)r' < infinity} onstruct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to <(L)over cap>(r)-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrodinger equation. (C) 2017 Elsevier Masson SAS. All rights reserved.
  • Satoshi Masaki, Jun-Ichi Segata
    SIAM JOURNAL ON MATHEMATICAL ANALYSIS 50 (3) 2839 - 2866 0036-1410 2018 [Refereed]
     
    In this paper, we give an improvement of nondiagonal Strichartz estimates for the Airy equation by using a Morrey-type space. As its applications, we prove the small data scattering and the existence of special nonscattering solutions, which are minimal in a suitable sense, to the mass-subcritical generalized Korteweg-de Vries equation. Especially, the use of a re fi ned nondiagonal estimate removes several technical restrictions on the previous work [S. Masaki and J. Segata, Existence of a Minimal Non-Scattering Solution to the Mass-Subcritical Generalized Korteweg-de Vries Equation, preprint, arXiv:1602.05331] about the existence of the special non-scattering solution.
  • Satoshi Masaki, Hayato Miyazaki
    SIAM JOURNAL ON MATHEMATICAL ANALYSIS 50 (3) 3251 - 3270 0036-1410 2018 [Refereed]
     
    In this paper, we consider the final state problem for the nonlinear Schrodinger 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 nonpolynomial nonlinearity, we decompose it into a resonant part and a nonresonant 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 nonresonant part is an infinite sum in general.
  • Rowan Killip, Satoshi Masaki, Jason Murphy, Monica Visan
    NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS 24 (4) 1021-9722 2017/08 [Refereed]
     
    We consider the mass-subcritical nonlinear Schrodinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity s(c) is an element of (max{-1, -d/2}, 0), we prove that any solution satisfyin on its maximal interval of existence must be global and scatter.
  • Satoshi Masaki
    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 42 (4) 626 - 653 0360-5302 2017 [Refereed]
     
    We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in a weighted L-2 space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all nonscattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing wave solution.
  • Satoshi Masaki, Jun-Ichi Segata
    ANALYSIS & PDE 9 (3) 699 - 725 1948-206X 2016 [Refereed]
     
    The purpose of this paper is to study local and global well-posedness of the initial value problem for the generalized Korteweg-de Vries (gKdV) equation in (L) over cap (r) = {f is an element of S' (R) : parallel to f parallel to((L) over capr) = parallel to (f) over cap parallel to(Lr') < infinity}. We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical <(L)over cap>(r)-space. A key ingredient is a Stein-Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for (L) over cap (r)-framework.
  • Satoshi Masaki, Takayoshi Ogawa
    JOURNAL OF MATHEMATICAL PHYSICS 56 (12) 0022-2488 2015/12 [Refereed]
     
    In this paper, we study a dispersive Euler-Poisson system in two dimensional Euclidean space. Our aim is to show unique existence and the zero-dispersion limit of the time-local weak solution. Since one may not use dispersive structure in the zero-dispersion limit, when reducing the regularity, lack of critical embedding H-1 subset of L-infinity becomes a bottleneck. We hence employ an estimate on the best constant of the Gagliardo-Nirenberg inequality. By this argument, a reasonable convergence rate for the zero-dispersion limit is deduced with a slight loss. We also consider the semiclassical limit problem of the Schrodinger-Poisson system in two dimensions. (C) 2015 AIP Publishing LLC.
  • Satoshi Masaki
    COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 14 (4) 1481 - 1531 1534-0392 2015/07 [Refereed]
     
    This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schrodinger equation of power type with data in a critical homogeneous weighted L-2 space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted L-2 space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previous author's result to the case where the exponent of nonlinearity is below so-called Strauss number. A main new ingredient is a stability estimate in a Lorenz-modified-Bezov type spacetime norm.
  • Masaya Maeda, Satoshi Masaki
    NONLINEAR DYNAMICS IN PARTIAL DIFFERENTIAL EQUATIONS 64 273 - 280 2015 [Refereed]
     
    We consider nonlinear Schrodinger equation with nonlocal nonlinearity which is described by a growing interaction potential. This model contains low-dimensional Schrodinger Poisson system. We briefly survey recent progress on this subject and then show existence of ground state in a specific model.
  • Masaya Maeda, Satoshi Masaki
    DIFFERENTIAL AND INTEGRAL EQUATIONS 26 (7-8) 731 - 756 0893-4983 2013/07 [Refereed]
     
    In this article, we consider the nonlinear Schrodinger equation with nonlocal nonlinearity, which is a generalized model of the Schrodinger-Poisson system (Schrodinger-Newton equations) in low dimensions. We prove global well-posedness in a wider space than in previous results and show the stability of standing waves including excited states. It turns out that an example of stable excited states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.
  • Satoshi Masaki
    JOURNAL OF DIFFERENTIAL EQUATIONS 251 (11) 3028 - 3062 0022-0396 2011/12 [Refereed]
     
    We consider the semiclassical Schrodinger-Poisson system with a special initial data of WKB type such that the solution of the limiting hydrodynamical equation becomes time-global in dimensions at least three. We give an example of such initial data in the focusing case via the analysis of the compressible Euler-Poisson equations. This example is a large data with radial symmetry, and is beyond the reach of the previous results because the phase part decays too slowly. Extending previous results in this direction, we justify the WKB approximation of the solution with this data for an arbitrarily large interval of R(+). (C) 2011 Elsevier Inc. All rights reserved.
  • Satoshi Masaki
    SIAM JOURNAL ON MATHEMATICAL ANALYSIS 43 (6) 2719 - 2731 0036-1410 2011 [Refereed]
     
    We consider the Cauchy problem of the two-dimensional Schrodinger-Poisson system in the energy class. Though the Newtonian potential diverges at spatial infinity in the logarithmic order, global well-posedness is proven in both defocusing and focusing cases. The key is a decomposition of the nonlinearity into a sum of the linear logarithmic potential and a good remainder, which enables us to apply the perturbation method. Our argument can be adapted to the one-dimensional problem.
  • Satoshi Masaki
    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 35 (12) 2253 - 2278 0360-5302 2010 [Refereed]
     
    We consider the Schrodinger-Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schrodinger equation in the semiclassical limit.
  • Satoshi Masaki
    KYUSHU JOURNAL OF MATHEMATICS 63 (1) 51 - 82 1340-6116 2009/03 [Refereed]
     
    We study an asymptotic expansion near t = infinity of the solution to the Cauchy problem for the nonlinear Schrodinger equation with repulsive short-range nonlinearity of power type We construct two kinds of approximate solution with asymptotic expansions The first is an accurate approximate solution and of abstract form The second is the approximation of the first and of explicit form The sharpness of these approximations strongly depend on the fractional part of the power of the nonlinearity In particular, if the power is an integer, we obtain a complete expansion of the solution
  • Remi Carles, Satoshi Masaki
    ASYMPTOTIC ANALYSIS 58 (4) 211 - 227 0921-7134 2008 [Refereed]
     
    We justify WKB analysis for Hartree equation in space dimension at least three, in a regime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrodinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.
  • Satoshi Masaki
    ANNALES HENRI POINCARE 8 (6) 1037 - 1069 1424-0637 2007/09 [Refereed]
     
    We study the semi-classical limit of the Hartree equation, which has focusing at a point. There exists a critical index indicating nonlinear effect around the caustic, and it is known that the influence by the nonlinearity is negligible in subcritical case (called linear caustic case), and that it is not in critical case (nonlinear caustic case). We give the asymptotic behavior beyond caustic in some supercritical cases which give rise to very strong nonlinear effect.

MISC

Research Projects

  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2024/04 -2028/03 
    Author : 眞崎 聡, 水谷 治哉, 瓜屋 航太, 山崎 陽平
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 2021/04 -2026/03 
    Author : 三浦 英之, 眞崎 聡, 前川 泰則
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2021/04 -2025/03 
    Author : 瀬片 純市, 若狭 徹, 眞崎 聡, 高田 了, 山崎 陽平
  • 日本学術振興会：科学研究費助成事業 基盤研究(B)
    Date (from‐to) : 2017/04 -2022/03 
    Author : 中西 賢次, 水谷 治哉, 眞崎 聡
     
    眞崎は質量劣臨界の非線形 Schrodinger 方程式について、球対称で負の斉次 Sobolev 空間に入る解を調べた。反発性非線形項の場合は、臨界ノルムが有界な大域解の漸近挙動は散乱で与えられることを示し、集約性の場合は、臨界ノルムの大域上界が最小となる非散乱解を構成し、その解軌道がプレコンパクトであることを示した。また、吸引的なデルタポテンシャルを持つ１次元非線形 Schrodinger 方程式について、小さい初期値に対するソリトン分解定理、即ち時刻無限大でソリトンと散乱波の和に漸近することを示した。 水谷はポテンシャル付き非線形 Schrodinger 方程式のエネルギー散乱への応用を念頭に、線形散乱において (修正) 波動作用素が L2 上で存在する場合に Sobolev 空間上でも存在するための十分条件を抽象的な枠組で導出した。例えば１階の Sobolev 空間の場合、この条件は臨界特異性を持つ短距離型、滑らかな長距離型、一次元点相互作用等のポテンシャルを含み、更に Schrodinger 方程式以外の分散型方程式にも適用できる。 中西は４次元 Zakharov 系に対する大域ダイナミクスの結果を球対称から一般解へ拡張する為、大域 Strichartz 評価の証明枠組を解作用素展開とプロファイル分解の観点から簡素化し、また鍵となる双線形評価を得た。量子効果付き Zakharov 系に対しては L2 解の初期値問題を考察し、局所・大域適切性および散乱について、成立する空間次元に関して古典系より大幅に改善されることを示した。他方、２次元のエネルギー臨界である二乗指数型非線形項の熱方程式に対して初期値問題の強解一意性の崩れを示し、関連する Trudinger-Moser 不等式に対して、最大化元の存在・非存在を分ける非線形項の境界を、第３項まで具体的な漸近展開で与えた。
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    Date (from‐to) : 2017/04 -2021/03 
    Author : 瀬片 純市, 眞崎 聡, 前田 昌也, 高田 了, 生駒 典久
     
    本研究課題では物理学, 工学に現れる非線形分散型方程式に対し, ソリトンおよび散乱という立場から研究を行っている. 研究代表者の瀬片は, gauge不変な非線形項をもつKlein-Gordon方程式の複素数値解の時刻無限大での詳細な挙動を, 解の漸近形がみたす常微分方程式を精密に解析することにより捉える事ができた. また, Jason Murphy氏, 研究分担者の眞崎氏とともに, 前年度に引き続き, 吸引的なデルタポテンシャルをもつ非線形シュレディンガー方程式のソリトンのまわりでの解の長時間挙動について研究を行った. これまでは小さなソリトンのまわりの解について考察してきたが, 今年度は, 線形の散乱理論を援用することで, 必ずしも小さいとは限らないソリトンのまわりの解の挙動について考察した. 研究分担者の前田は, Scipio Cuccagna氏とともに吸引的なデルタポテンシャルをもつ非線形シュレディンガー方程式に対し, virial型の議論をすることで質量劣臨界の場合に小さな解が時刻無限大でソリトンと分散波に分かれることを証明した. また, 非線形シュレディンガー方程式の臨界周波数付近でのソリトン解の振動を解析した. 研究分担者の高田は, 2次元非粘性成層 Boussinesq方程式の初期値問題を考察し, 最適な初期正則性のもとで長時間可解性を証明した. 研究分担者の生駒は, 質量が一定という制約条件の下，ハミルトニアンを最小化するにする関数の存在および非存在を考察した. 特に調和ポテンシャルのように強い効果を持たないポテンシャル関数と一般的な非線形項の取り扱いに成功した. また, 2つの冪乗型非線形項を持つ非線形シュレディンガー方程式に対する基底状態解の一意性および非退化性を示した. 特に1つの冪はSobolev臨界であり，周波数が非常に大きい状態を取り扱った.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)
    Date (from‐to) : 2017/04 -2021/03 
    Author : Masaki Satoshi
     
    We study the large-time behavior of solutions to the nonlinear dispersive equations. The biggest contribution is the classification of the global behavior of solutions to the nonlinear Schrodinger equation with linear potential below the first excited energy. As preliminary studies, we consider the delta potential case and obtain the modified scattering and the asymptotic stability of solitons. By using the knowledge obtained in these studies, the above result is obtained. We also obtained results on modified scattering. We had much more progress than expected. We obtain the modified scattering result for equations with a general homogeneous critical nonlinearity and the classification result for cubic dispersive systems in one space dimension.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))
    Date (from‐to) : 2019 -2021 
    Author : Masaki Satoshi
     
    We study the large-time behavior of solutions to the nonlinear dispersive equations. The biggest contribution is the classification of the global behavior of solutions to the nonlinear Schrodinger equation with linear potential below the first excited energy. As preliminary studies, we consider the delta potential case and obtain the asymptotic stability of solitons. By using the knowledge obtained in this study, the above result is obtained. We also obtained results on modified scattering. We had much more progress than expected. We establish the classification result for cubic dispersive systems in one space dimension. This enables us to a systematic approach to understand the whole picture of the behavior for systems.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)
    Date (from‐to) : 2012/04 -2016/03 
    Author : Masaki Satoshi
     
    The first intend of the research was to focus on quantum-fluid properties of Schrodinger equations and to clarify the correspondence between blowup of solution to a nonlinear Schrodinger equation and collapse of solution of a corresponding classical fluid equation. The intended method for this analysis is a contradiction argument that construct a virtual solution from failure of a target conclusion. In this decade, there is much progress on this kind of argument. The research progresses unexpectedly and a special non-scattering solution which is minimal in a suitable sense is found for mass-subcritical equations. Behavior of the solution is completely unknown and different from any of those we knew before. I also consider generalized KdV equations and find that similar kind of threshold solution exists.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Research Activity Start-up
    Date (from‐to) : 2010 -2011 
    Author : MASAKI Satoshi
     
    Two dimensional version of Schrodinger-Poisson system, which is a model equation for semiconductor devices, has a nonlinear potential growing at the spatial infinity due to the fact that the Newtonian kernel, which is a fundamental solution of the Poisson equation in two dimensions, becomes a logarithmic function. Our research concerns Schrodinger equations with a nonlocal nonlinearity given by more general kernels growing at the spatial infinity. Although a derivation is rather simple, this type of equations requires quite different mathematical treatment. We first establish a way to treat this class of nonlinear Schrodinger equations rigorously by introducing a novel transform of equation which is based on conservative quantities of Schrodinger equations. In particular, it turns out that nonlocal nonlinearities of this type contain an effect like a linear potential. Based on this fact, we are able to prove time-global well-posedness result in an energy class. Moreover, when integral kernel is a quadratic function, it turns out that the solutions are written explicitly nevertheless the equation is fully nonlinear. The behavior of this explicit solution, which can be analyzed completely, helps up to understand the effects of nonlinearities which we concern. This example also reveals that there exists a stable excited state.
  • 日本学術振興会：科学研究費助成事業 特別研究員奨励費
    Date (from‐to) : 2009 -2011 
    Author : 眞崎 聡
     
    非線型シュレディンガー方程式にプランク定数に相当するパラメータをいれて、そのパラメータをゼロに近づける半古典曲極限と呼ばれる極限下での解の挙動を探る。この極限は物理現象をマクロな視点から見ていることに対応しており、量子力学で支配される世界から古典力学で支配される世界への移行を記述する。 21年度の目標として、流体力学の手法を学び、古典軌道を解析することを挙げた。これについて大きな成果が得られた。特に、シュレディンガー・ポアッソン方程式系において、ある特殊な形の初期値を与えると、対応する古典軌道が焦点を形成せずに時刻無限大まで伸びることがわかった。この研究では対応するオイラー・ポアッソン方程式の解の非常に詳細な解析が必要であった。また、この特殊例に対しては、半古典極限におけるWKB型の解の近似が準大域的に成立することも示した。 また、6月にフランス・イギリスに一か月滞在して、国際研究集会において研究発表を行い、また海外の研究者と議論を交わし情報交換を行った。そこでの情報交換をもとにして、シュレディンガー・ポアッソン方程式系における解の半古典極限におけるWKB型近似を空間2次元に対して拡張した。先行結果で1次元と3次元以上に関しては知られていたが、2次元についてはわかっていなかった。 この結果に関しては反響も大きく、9月に開催された日本数学会2009年度総合分科会で初めて発表したのち、約3カ月の間にセミナーや研究集会などで7回もの講演の機会を得ることにつながった。
  • 日本学術振興会：科学研究費助成事業 特別研究員奨励費
    Date (from‐to) : 2007 -2008 
    Author : 眞崎 聡
     
    本研究の目標は,半古典パラメータを持つ非線型Schrodinger方程式において,焦点を持つような初期値を与え,半古典パラメータがOに近づいた際の解の漸近挙動が非線型項の「強さ」や「減衰度」(非線型項の短距離性・長距離性にかかわる性質)に応じてどう変化するを調べ,非線型Schrodinger方程式において焦点が存在することがどのような影響を及ぼすのか探ることであった. 20年度では,べき乗型の非線型項に対する先行研究をHartree方程式に対しても拡張し,さらに以下の二つの点で改良した.第一の点は,取り扱う非線型項の「強さ」に関する制限を緩めたこと.第二の点は動く焦点の形成を示したことである.この研究は論文にまとめ,現在投稿中である.また,上で述べた研究の第二の点に関わる"古典軌道"の解析に大きな進展があった.ここでいう"古典軌道"とは考察する非線型Schrodinger方程式を"量子力学を記述する方程式"とみなした時に,対応する"古典力学を記述する方程式"の解のことである.この対応する方程式とは圧縮性のEuler方程式になる.Euler方程式の解は有限時間で崩壊してしまうことがあり,一般には大域的ではない.この解か有限時間で崩壊することが本研究のターゲットである焦点の形成と対応している.球対称な場合に圧縮性のEuler-Poisson方程式に対してその古典解か大域的になる必要十分条件を導いた.この結果はRemarks on global existence of classical solution to multi-dimensional compressible Euler-Poisson equations with geometrical symmetry, RIMS Kokyuroku Bessatsuとして発表した.

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