Long Time Existence for a Strongly Dispersive Boussinesq System
Jean‐Claude Saut, Li Xu
Abstract
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.