Collisions of localized patterns in a nonvariational Swift-Hohenberg equation
Collisions of localized patterns in a nonvariational Swift-Hohenberg equation
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非变分 Swift-Hohenberg 方程中局部模式的碰撞
DOI：
10.1103/physreve.107.064214
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发表时间：
2023
影响因子：
2.4
通讯作者：
Knobloch, Edgar
中科院分区：
文献类型：
--
作者：
Raja, Mathi;van Kan, Adrian;Foster, Benjamin;Knobloch, Edgar
三次五次 Swift-Hohenberg 方程 (SH35) 已被提出作为层中面具有反射对称性的几个对流系统（包括二元流体对流）的阶次参数描述。我们使用数值延拓以及广泛的直接数值模拟 (DNS) 来研究 SH35，并使用附加的非变分二次项来模拟打破中面反射对称性的影响。模型的非变分结构导致不对称空间局部结构（LS）的传播。在弱对称破缺极限下导出的此类结构漂移速度的渐近预测已得到数值验证。接下来，我们对相同和不同的移动结构之间可能发生的碰撞场景进行了广泛的研究，改变了类似温度的控制参数。这些碰撞是非弹性的，会导致结构静止或移动。根据系统参数和碰撞结构的类型，最终状态可能是初始 LS 的简单束缚态，但由于非线性相互作用，它也可能比两个初始状态的总和更长或更短。变分系统的麦克斯韦点（其中全局模式状态的自由能等于平凡状态的自由能）被证明与实现这些场景中的哪一个无关。相反，我们认为束缚态的稳定性是关键。虽然各个 LS 位于非变分 SH35 中改进的蛇梯结构上，但碰撞产生的多脉冲束缚态位于参数空间中的孤岛上，与平凡解无关。在梯度 SH35 中，此类孤岛始终为 8 字形形状，但在目前的非梯度情况下，它们通常更复杂，尽管在一小部分情况下保留了 8 字形形状。其中一些复杂的孤岛显示终止于 T 点分叉。提出了一个简化模型来描述 LS 尾部之间的相互作用。该模型由两个耦合常微分方程 (ODE) 组成，捕获线性水平上 SH35 剖面的振荡性质。它包含三个参数：两个相互作用幅度和一个相位，其值是使用梯度下降优化从高分辨率 DNS 推导出来的。对于导致形成简单束缚态的碰撞，简化模型以高定量精度再现了 LS 的轨迹。当非线性相互作用导致波长的创建或删除时，模型的性能较差。最后，我们根据相对于自由传播的净吸引力或排斥力提出给定相互作用的有效特征。研究发现，无论两个最接近的相互作用极值是否具有相同或相反的符号，网络中的相互作用可以是吸引的或排斥的。我们的研究结果强调了这种双稳态非变分 SH35 所描述的丰富的时间动态，并表明该系统中的相互作用可以在很大程度上通过高度简化的 ODE 模型定量捕获。
The cubic-quintic Swift-Hohenberg equation (SH35) has been proposed as an order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use numerical continuation, together with extensive direct numerical simulations (DNSs), to study SH35 with an additional nonvariational quadratic term to model the effects of breaking the midplane reflection symmetry. The nonvariational structure of the model leads to the propagation of asymmetric spatially localized structures (LSs). An asymptotic prediction for the drift velocity of such structures, derived in the limit of weak symmetry breaking, is validated numerically. Next, we present an extensive study of possible collision scenarios between identical and nonidentical traveling structures, varying a temperaturelike control parameter. These collisions are inelastic and result in stationary or traveling structures. Depending on system parameters and the types of structures colliding, the final state may be a simple bound state of the initial LSs, but it can also be longer or shorter than the sum of the two initial states as a result of nonlinear interactions. The Maxwell point of the variational system, where the free energy of the global pattern state equals that of the trivial state, is shown to have no bearing on which of these scenarios is realized. Instead, we argue that the stability properties of bound states are key. While individual LSs lie on a modified snakes-and-ladders structure in the nonvariational SH35, the multipulse bound states resulting from collisions lie on isolas in parameter space, disconnected from the trivial solution. In the gradient SH35, such isolas are always of figure-eight shape, but in the present nongradient case they are generically more complex, although the figure-eight shape is preserved in a small subset of cases. Some of these complex isolas are shown to terminate in T-point bifurcations. A reduced model is proposed to describe the interactions between the tails of the LSs. The model consists of two coupled ordinary differential equations (ODEs) capturing the oscillatory nature of SH35 profiles at the linear level. It contains three parameters: two interaction amplitudes and a phase, whose values are deduced from high-resolution DNSs using gradient descent optimization. For collisions leading to the formation of simple bound states, the reduced model reproduces the trajectories of LSs with high quantitative accuracy. When nonlinear interactions lead to the creation or deletion of wavelengths, the model performs less well. Finally, we propose an effective signature of a given interaction in terms of net attraction or repulsion relative to free propagation. It is found that interactions can be attractive or repulsive in the net, irrespective of whether the two closest interacting extrema are of the same or opposite signs. Our findings highlight the rich temporal dynamics described by this bistable nonvariational SH35, and show that the interactions in this system can be quantitatively captured, to a significant extent, by a highly reduced ODE model.