耦合Boussinesq方程和一族Ostrovsky方程的初值问题
Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations
摘要：
我们考虑的初始值问题耦合Boussinesq方程系统的无限线上的本地化或足够迅速衰减的初始数据，产生足够迅速衰减的右和左传播波。研究弱非线性波的动力学，利用渐近多尺度展开和对快时间的平均，得到了一族渐近精确的耦合和非耦合单向波Ostrovsky方程.然后，我们构造了一个弱非线性解的初值问题中导出的Ostrovsky方程的控制方程的精度内的解决方案，并表明，有没有长期条款。当耦合参数为零时，我们的结果得到了Boussinesq方程初值问题的弱非线性解，其形式为两个Korteweg-de Vries方程的初值问题的解，可通过逆散射变换积分.我们还进行了相关的数值模拟的原始未近似系统的Boussinesq方程，以说明其解决方案的行为不同的渐近制度的差异。
英文摘要：
We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scales expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.
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