Supplementary information files for: Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction

Supplementary information files for: Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction
Coupled Boussinesq equations are used to describe long weakly-nonlinear longitudinal strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft adhesive). From the mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations, for the right- and left-propagating waves in each layer. However, the models impose a “zero-mass constraint” i.e. the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow-time variables. By construction, the Ostrovsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behaviour of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalised) conservation laws of the coupled Boussinesq equations for strains are derived and used to control the accuracy of the numerical simulations.

附加信息文件：消除零质量矛盾的耦合Boussinesq方程及Ostrovsky型模型的周期解
耦合Boussinesq方程被用于描述双层介质中长波弱非线性纵向应变波，其中层间存在软性结合（例如软性粘合剂）。从数学角度而言，当层间线性长波速度存在显著差异时（高对比情形），将呈现出尤为棘手的情形。传统单向模型推导过程中，针对每一层中右向和左向传播的波，导出四个独立的Ostrovsky方程。然而，这些模型施加了一个‘零质量约束’，即初始条件必然要求具有零均值，从而限制了该描述的应用范围。在本研究中，我们通过构建偏离演化平均值的解法，绕过该高对比情形下的矛盾，该解法涉及包含两对快速特征变量和两个慢时间变量的渐近多尺度展开。通过构造，本推导范围内的Ostrovsky方程求解的是具有零均值的初始条件，而原系统的初始条件可能具有非零均值。我们通过对解的渐近有效性进行细致的数值检验。我们将模型应用于描述由孤立波初始条件产生的相向传播波，或由孤波初始条件产生的共向传播波，以及由此产生的波相互作用，并与层间线性长波速度接近（低对比情形）的双层介质中波的行为进行对比。对于应变耦合Boussinesq方程，推导了一对局部（经典）和两对非局部（广义）守恒定律，并用于控制数值模拟的精度。