自发运动阈值附近化学活性粒子的弱非线性动力学。I.伴随方法
Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. I. Adjoint method
摘要：
在这个系列中，我们研究了自发运动阈值附近的化学活性粒子的弱非线性动力学。在这一部分中，我们专注于稳定的解决方案，并制定了一个“伴随方法”推导出的非线性振幅方程的粒子的速度，首先假设在文献中的各向同性化学活性粒子的正则模型，然后考虑一般的扰动模型。在以前的作品中，振幅方程是从一个可解性条件上的非齐次问题，在第二阶的粒子尺度弱非线性膨胀，制定该问题涉及渐近匹配的领先的解决方案，在一个偏远地区的平流和扩散是平衡的。我们开发了一个广义的可解性条件的基础上Fredholm替代参数，这需要确定伴随线性算子的阈值和计算其内核。这就避免了在早期理论中显然需要解决二阶非齐次问题的问题，从而大大简化了问题，并通过在一个共同的基础上处理各种各样的扰动情况来增加洞察力。为了说明我们的方法，我们推导和求解振幅方程的扰动情况下（外力和扭矩场，非均匀的表面性质，一阶表面动力学和散装吸收），表明足够接近阈值弱扰动可以明显修改和丰富的景观稳定的解决方案。
英文摘要：
In this Series, we study the weakly nonlinear dynamics of chemically active particles near the threshold for spontaneous motion. In this part, we focus on steady solutions and develop an `adjoint method' for deriving the nonlinear amplitude equation governing the particle's velocity, first assuming the canonical model in the literature of an isotropic chemically active particle and then considering general perturbations about that model. As in previous works, the amplitude equation is obtained from a solvability condition on the inhomogeneous problem at second order of a particle-scale weakly nonlinear expansion, the formulation of that problem involving asymptotic matching with a leading-order solution in a remote region where advection and diffusion are balanced. We develop a generalised solvability condition based on a Fredholm Alternative argument, which entails identifying the adjoint linear operator at the threshold and calculating its kernel. This circumvents the apparent need in earlier theories to solve the second-order inhomogeneous problem, resulting in considerable simplification and adding insight by making it possible to treat a wide range of perturbation scenarios on a common basis. To illustrate our approach, we derive and solve amplitude equations for a number of perturbation scenarios (external force and torque fields, non-uniform surface properties, first-order surface kinetics and bulk absorption), demonstrating that sufficiently near the threshold weak perturbations can appreciably modify and enrich the landscape of steady solutions.
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