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 日程： 2015年7月27日(月)，28日(火)
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場所： 東京大学大学院数理科学研究科棟 002教室 （京王井の頭線駒場東大前駅よりすぐ）
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講師（講演順，敬称略）：
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 プログラム：
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1. Frank Merle 氏： "Asymptotic behavior of dispersive problems and solitons"
Part I : Construction of type II blow-up solution for supercritical NLS
We consider the energy super critical nonlinear Schrödinger equation
in large dimensions d≧11 with spherically symmetric data. For all p>p(d) large enough, in particular in the super critical regime:
s_c = d/2 - 2/(p-1) > 1,
we construct a family of C^∞ finite time blow up solutions which become singular via concentration of a universal profile
with the so called type II quantized blow up rates:
The essential feature of these solutions is that all norms below scaling remain bounded
Our analysis fully revisits the construction of type II blow up solutions for the corresponding heat equation which was done using maximum principle techniques following [Matano-Merle (2004)].
Instead we develop a robust energy method, in continuation of the works in the energy critical case and the L2 critical case.
This shades a new light on the essential role played by the solitary wave and its tail in the type II blow up mechanism, and the universality of the corresponding singularity formation in both energy critical and super critical regimes.
講演資料： Merle-I
Part II : Compactness/dispersion principle for dispersive equations, applications for asymptotic profiles
1) We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave,
and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5.
This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time.
2) We then consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution.
We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves.
This result is a consequence of a new general compactness/rigidity argument based on profile decomposition.
We also give an application of this method to the energy-critical Schrödinger equation.
3) We proved with an additional non-degeneracy assumption that the only solutions with the compactness property are stationary solutions and solitary waves that are Lorentz transforms of the former.
講演資料： Merle-II-1 Merle-II-2 Merle-II-3
２．中西賢次 (Kenji Nakanishi) 氏： "Global behavior of large solutions for NLS with potential"
Abstract：
Nonlinear dispersive equations generate solutions with various space-time behavior.
The three major types are: scattering, blow-up, and solitons, which are further classified into stable and unstable ones.
Stable solitons are important by themselves, corresponding to some characteristic physical phenomena.
Unstable ones are also important to understand the global dynamics of the equation, as they often appear on the threshold between different types of solutions.
In general, a single equation or system can have a few stable solitons and many unstable solitons,
but the global dynamics containing both of them are yet to be explored from the mathematical or PDE viewpoint,
especially when solitons are large and distinct from each other.
In this talk, we investigate a simple model case, which is the nonlinear Schrödinger equation with a potential
and a focusing cubic nonlinearity, assuming that the linear equation with the potential has a single negative eigenvalue.
The nonlinear equation has small stable solitons due to the potential, and large unstable solitons due to the nonlinearity.
Restriction to small mass (L^2 norm) enables us to characterize the solitons, at least for the lowest
energy (the ground state) and the second lowest energy (the first excited state).
Then we consider the first natural question: Classify the global dynamics in terms of the initial data in an
energy region which contains only the ground state and the first excited state.
非線形分散型方程式は様々な時空挙動の解を持つ。三つの典型は散乱、爆発、ソリトンで、ソリトンは更に安定なものと不安定なものに分類される。安定なソリトンは、それ自体が特徴的な物理現象を表すものとして重要である。不安定なソリトンも、しばしば異なる解の境に現れるため、方程式の大域挙動を理解するために重要である。
一般に、一つの方程式またはシステムが幾つかの安定なソリトンと多数の不安定なソリトンを持ち得るが、両者を含む大域ダイナミクスは、特にソリトンが大きく互いに異なる場合、数学的(PDEの)観点からは未だ良く分かっていない。
この講演では簡単なモデルケースとして、ポテンシャルと集約的３次の非線形項を持つ非線形シュレディンガー方程式で、ポテンシャル付の線形方程式が単独の負の固有値を持つ場合について調べる。この非線形方程式はポテンシャルから来る小さな安定ソリトンと、非線形項から来る大きな不安定ソリトンを持つ。小さな質量(L^2ノルム)に制限すれば、少なくとも最小エネルギー(基底状態)と２番目のエネルギー(第１基底状態)についてはソリトンを特徴付けられる。そこで考えるのは最初の自然な問題である：基底状態と第１励起状態のみが入るようにエネルギー領域を制限して大域ダイナミクスを初期状態から分類せよ。
アブストラクトPDF版： Nakanishi
３. 前田 昌也 (Masaya Maeda) 氏： "Stabilization of small solutions for NLS with potential"
Abstract：
In this talk, we consider the long time behavior of small solutions of cubic nonlinear Schrödinger equations with potential in 3D:
We assume that the Schrödinger operator -Δ + V have more than two simple negative eigenvalues. By spectral decomposition, it is easy to see that the solution of linear Schrödinger equation with potential decouples in quasi-periodic solutions (sum of periodic solutions corresponding to the eigenvalues of Schrödinger operator) and scattering waves (which corresponds to the continuous spectrum of the Schrödinger operator).
For the nonlinear case, naively one can think that the dynamics is similar to the linear case because we are considering small solutions (at least short time). Further, it is known that there exists small solitary waves (or soliton in short) which bifurcate from the eigenvalues of the Schrödinger operator. However, it is shown that under suitable assumptions, all small solutions decouples into one soliton and scattering wave and in particular there exists no quasi-periodic solutions.
In this talk, I will try to explain the mechanism which prohibits the coexistence of two solitons. This is a nonliear interaction (called Fermi Golden Rule) between the discrete spectrum and continuous spectrum which creates radiation damping. In addition, if I have time, I will talk about discrete nonlinear Schrödinger equations with potential which admits two mode quasi-periodic solutions. This fact is due to the absence of Fermi Golden Rule.
本講演では線形ポテンシャルをもつ3次冪非線形シュレディンガー方程式の小さな解の時間大域挙動を考察する。ここではシュレディンガー作用素は2つ以上の固有値をもつと仮定する。
まず、ポテンシャルをもつ線形シュレディンガー方程式の解はシュレディンガー作用素のスペクトル分解により解の様子がよくわかる。実際、線形の場合では解は固有値成分に対応する空間的に局在した時間周期解の和と連続スペクトル成分に対応する散乱解に分解される。
非線形の場合でもここでは小さな解を考えているためその挙動は（少なくとも短い時間では）線形の解に近いと考えられる。また、分岐により非線形シュレディンガー方程式は小さな定在波解（小さなソリトン）をもつことがわかる。
しかし、長時間の挙動では非線形相互作用により非線形方程式の解の挙動は線形の場合と真に異なる。実際、非線形シュレディンガー方程式の小さな解は一つの小さなソリトンと散乱解に分解される。特に、二つ以上のソリトン解は共存できない。
この講演では二つ以上のソリトンが共存できない理由を、固有スペクトルと連続スペクトルの非線形相互作用の観点から説明したい。この非線形相互作用はフェルミの黄金律と呼ばれる。また、時間が許せば離散非線形シュレディンガー方程式の準周期解の存在についても話したい。
アブストラクトPDF版： Maeda
４. 水町 徹 (Tetsu Mizumachi) 氏： "Stability of line solitons for KP-II"
Abstract：
The KP-II equation
is a 2-dimensional generalization of the KdV equation that takes slow variations in the transversal direction into account. It describes the motion of shallow water waves with weak surface tension. Any solutions of the KdV equation formally satisfy the KP-II equation.
In this lecture, I will talk on the transversal stability of KdV 1-solitons as solutions of the KP-II equation (1-line solitons) and explain modulations of the line soliton solutions are described by a system of Burgers' equations.
The KP-II equation is a Hamiltonian system and in L^2(R^2), the spectrum of the linearized operator around the line solitons consists of the entire imaginary axis. However, in an exponentially weighted space where the size of perturbations are biased in the direction of motion, the spectrum of the linearized operator consists of a curve which goes through 0 and the set of continuous spectrum which locates in the stable half plane and is away from the imaginary axis. The former one appears because line solitons are not localized in the transversal direction and is related to the modulation of line solitons. Thanks to the finite speed propagation of modulations of the line soliton along its crest, the set of 1-line solitons is not stable in L^2(R^2) but stable in L^2_loc(R^2).
KP-II方程式は空間1次元の長波長近似モデルであるKdV方程式
に波の主な進行方向(x方向)と垂直な方向(y方向)の波の緩やかな変化を取れ入れたモデルであり2次元の水面波の運動を記述する．u(t,x) がKdV方程式の解であれば，y方向に一様なKP-II方程式の解になっている．
KdV方程式のの1-soliton解
をKP-II方程式のy方向に一様な進行波解(line soliton)とみなした場合の安定性について解説する．KP-II方程式はハミルトニアン系であり，エネルギーやL^2-ノルムなどの保存則をもつ．line solitonはL^2(R^2)に属する摂動に対して線形中立安定であるが，line solitonの進行方向に指数増大する関数を重み関数とする重み付空間においては，線形化作用素のスペクトルは0を通る曲線部分を除くと安定半平面の内側にある．0を通る曲線部分は，ソリトン解の周りでの線形化KdV作用素の0固有値が連続スペクトルに変化したものであり，line solitonの摂動による変調に関係している．line solitonの変調する様子はline solitonの速度と傾きに関するパラメータについての時間変数tと横断方向の変数yに関するBurgers方程式系で記述される．line solitonの変調は有限速度でline solitonの頂に沿って横方向に伝わるため，line soliton解はL^2_loc(R^2)で安定になる.
アブストラクトPDF版： Mizumachi
世話人： 俣野博（代表） 問い合わせ先： matano
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