1+1维标量场的扭结团簇动力学
Dynamics of kink clusters for scalar fields in dimension 1+1
摘要：
考虑1+1维实标量场方程，其偶正自相互作用势具有两个非简并零点（真空）1和-1。众所周知，这样的模型承认非平凡的静态解决方案称为扭结和antikinks。扭结群是一个解，在大的正时间内，它接近于速度收敛于0的交替扭结和反扭结的叠加。它们可以等价地表征为包含真空之间给定数量的跃迁的最小可能能量的解，或者其动能在很长时间内衰减到0的解。我们的主要结果是一个确定的主阶渐近行为的任何扭结集群。此外，我们构造了一个扭结集群的任何指定的初始位置的扭结和反扭结，只要它们的相互距离足够大。最后，我们表明，扭结集群的形成/崩溃的多扭结配置的通用配置文件。证据依赖于减少，使用适当选择的调制参数，一个有吸引力的指数相互作用的n体问题。
英文摘要：
We consider a real scalar field equation in dimension 1+1 with an even positive self-interaction potential having two non-degenerate zeros (vacua) 1 and -1. It is known that such a model admits non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to 0. They can be equivalently characterised as the solutions of minimal possible energy containing a given number of transitions between the vacua, or as the solutions whose kinetic energy decays to 0 for large time. Our main result is a determination of the main-order asymptotic behaviour of any kink cluster. Moreover, we construct a kink cluster for any prescribed initial positions of the kinks and antikinks, provided that their mutual distances are sufficiently large. Finally, we show that kink clusters are universal profiles for the formation/collapse of multi-kink configurations. The proofs rely on a reduction, using appropriately chosen modulation parameters, to an n-body problem with attractive exponential interactions.
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