SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2803-2848, January 2020.
This paper is concerned with the one-dimensional version of a specific member of the (abcd) family of Boussinesq systems having the higher possible dispersion and eigenvalues with nontrivial zeros. We will establish two different long time existence results for the solutions of the Cauchy problem. The first result concerns the system $\{ {\partial}_t\zeta+(1+\partial_x^2)\partial_xv+\partial_x(\zeta v)=0,\, \partial_tv+(1+\partial_x^2)\partial_x\zeta+\frac{1}{2}\partial_x(v^2)=0\}$ without a small parameter. If the initial data is of order $O(\varepsilon)$, we prove that the existence time scale is of $1/\varepsilon^{\frac{4}{3}}$, which improves the result $1/\varepsilon$ that could be obtained by a “dispersive" method (that is, using essentially the dispersive properties of the linear part). The second result is about the system $\{ \partial_t\zeta+(1+\epsilon\partial_x^2)\partial_xv+\epsilon\partial_x(\zeta v)=0,\, \partial_tv+(1+\epsilon\partial_x^2)\partial_x\zeta+\frac{\epsilon}{2}\partial_x(v^2)=0\} $, which involves a small parameter $\epsilon$ in front of the dispersive and nonlinear terms and which is the form obtained when the system is derived from the water wave system in the KdV/Boussinesq regime. If the initial data is of order $O(1)$, we obtain the existence time scale $1/{\epsilon^{\frac{2}{3}}}$, which improves the result $1/\sqrt{\epsilon}$ obtained by a dispersive method. These results were not included in the previous papers dealing with similar issues because of the presence of zeros in the phases. The proof involves normal form transformations suitably modified away from the zero set of the phases.

中文翻译：

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长时间存在的高度分散的Boussinesq系统

SIAM数学分析杂志，第52卷，第3期，第2803-2848页，2020年1月。
本文关注的是Boussinesq系统（abcd）族的特定成员的一维版本，其具有更高的离差和具有非平凡零点的特征值。对于柯西问题的解决方案，我们将建立两个不同的长期存在结果。第一个结果涉及系统$ \ {{\ partial} _t \ zeta +（1+ \ partial_x ^ 2）\ partial_xv + \ partial_x（\ zeta v）= 0，\，\ partial_tv +（1+ \ partial_x ^ 2）\ partial_x \ zeta + \ frac {1} {2} \ partial_x（v ^ 2）= 0 \} $（不带小参数）。如果初始数据的阶次为$ O（\ varepsilon）$，我们证明存在时间尺度为$ 1 / \ varepsilon ^ {\ frac {4} {3}} $，这将改善结果$ 1 / \ varepsilon $可以通过“分散”方法（即基本上使用线性零件的分散特性）获得。第二个结果是关于系统$ \ {\ partial_t \ zeta +（1+ \ epsilon \ partial_x ^ 2）\ partial_xv + \ epsilon \ partial_x（\ zeta v）= 0，\，\ partial_tv +（1+ \ epsilon \ partial_x ^ 2）\ partial_x \ zeta + \ frac {\ epsilon} {2} \ partial_x（v ^ 2）= 0 \} $，在色散和非线性项之前包含一个小参数$ \ epsilon $，其形式为当系统是从KdV / Boussinesq体制的水波系统派生而获得的。如果初始数据的阶次为$ O（1）$，我们将获得存在时间标度$ 1 / {\ epsilon ^ {\ frac {2} {3}}} $，这将改善结果$ 1 / \ sqrt {\ epsilon } $通过分散方法获得。由于阶段中存在零，因此这些结果未包含在处理类似问题的先前论文中。证明涉及从零相位集进行适当修改的正常形式转换。