分支结构和网络的非线性动力学
Nonlinear dynamics on branched structures and networks
摘要：
图上的非线性动力学已经迅速成为一个热门的问题与许多物理应用，从非线性光学到玻色-爱因斯坦凝聚。每当在物理实验中涉及到一个分枝结构时，用度量图或网络来近似这种结构是有用的。对于薛定谔方程来说，能量的非线性项中的六次方是临界的，因为在这个幂以下，约束能量是下界的，与质量的值无关（亚临界情况）。另一方面，如果非线性幂等于6，则下界取决于质量的值：低于临界质量，约束能量是下界的，超过它，它不是。对于大于6的幂次，约束能量泛函永远不会有下界，因此谈论基态（超临界情况）是没有意义的。这些结果与实直线上的非线性薛定谔方程的结果相同。事实上，关于基态的存在，图上系统的结果一般不同于线上系统的结果，即使在亚临界情况下也是如此：在后一种情况下，只要约束能量是下界的，总是存在基态（孤子，其形状是明确已知的），而对于图，基态的存在是不保证的。对于临界情况，我们的结果显示了一个现象比类似的线上丰富得多。
英文摘要：
Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging from nonlinear optics to Bose-Einstein condensation. Whenever in a physical experiment a ramified structure is involved, it can prove useful to approximate such a structure by a metric graph, or network. For the Schroedinger equation it turns out that the sixth power in the nonlinear term of the energy is critical in the sense that below that power the constrained energy is lower bounded irrespectively of the value of the mass (subcritical case). On the other hand, if the nonlinearity power equals six, then the lower boundedness depends on the value of the mass: below a critical mass, the constrained energy is lower bounded, beyond it, it is not. For powers larger than six the constrained energy functional is never lower bounded, so that it is meaningless to speak about ground states (supercritical case). These results are the same as in the case of the nonlinear Schrodinger equation on the real line. In fact, as regards the existence of ground states, the results for systems on graphs differ, in general, from the ones for systems on the line even in the subcritical case: in the latter case, whenever the constrained energy is lower bounded there always exist ground states (the solitons, whose shape is explicitly known), whereas for graphs the existence of a ground state is not guaranteed. For the critical case, our results show a phenomenology much richer than the analogous on the line.
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