考虑R^3 i \partial_t <$+ Δ <$+中的聚焦三次半线性Schroedinger方程| ψ| ^2 = 0。它承认一个八维流形的特殊解决方案称为基态孤子。我们在孤立子流形的邻近区域中展示了一个余维一临界实解析流形N，其具有渐进稳定的解。然后，我们表明，N是中心稳定的，在动力系统意义上的贝茨-琼斯，和全球的时间不变。N中的解是渐近稳定的，并且分离成两个渐近自由的部分，在极限下解耦-孤子和辐射。相反，在一般情况下，任何一直靠近孤子流形的解都在N中。证明采用调制的方法。新的元素包括一个不同的线性化和端点的时间依赖的线性化方程的Eschhartz估计。证明还使用了线性化的哈密顿量没有非零实本征值或共振的事实。这一点最近在这里所讨论的情形--R^3中的聚焦立方NLS--中由马左拉-辛普森和科斯汀-黄-施拉格的工作得到了证实。
英文摘要：
Consider the focusing cubic semilinear Schroedinger equation in R^3 i \partial_t ψ+ Δψ+ | ψ|^2 ψ= 0. It admits an eight-dimensional manifold of special solutions called ground state solitons.
We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant.
Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a soliton and radiation. Conversely, in a general setting, any solution that stays close to the soliton manifold for all time is in N.
The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time-dependent linearized equation.
The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here --- of the focusing cubic NLS in R^3 --- by the work of Marzuola-Simpson and Costin-Huang-Schlag.
In this paper we study Strichartz estimates for the half wave, the half Klein-Gordon and the Dirac Equations on compact manifolds without boundary, proving in particular for each of these flows local in time estimates both for the wave and Schroedinger admissible couples (in this latter case with an additional loss of regularity). The strategy for the proof is based on a refined version of the WKB approximation.
Consider the focussing cubic nonlinear Schrödinger equation in $R^3$: $$ iψ_t+Δψ= -|ψ|^2 ψ. $$ It admits special solutions of the form $e^{itα}ϕ$, where $ϕ$ is a Schwartz function and a positive ($ϕ>0$) solution of $$ -Δϕ+ αϕ= ϕ^3. $$ The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form $e^{i(v \cdot + Γ)} ϕ(\cdot - y, α)$. We prove that any solution starting sufficiently close to a standing wave in the $Σ= W^{1, 2}(…
Focusing on multi-solitons for the Klein-Gordon equations, in first part of this paper, we establish their conditional asymptotic stability. In the second part of this paper, we classify pure multi-solitons which are solutions converging to multi-solitons in the energy space as $t\rightarrow\infty$. Using Strichartz estimates developed in our earlier work \cite{CJ2} and the modulation techniques, we show that if a solution stays close to the multi-soliton family, then it scatters to the multi-soliton family in the sense that the solution will converge in large time to a superposition of Lorent…
