The Cauchy problem for the improved Boussinesq equation with spatially quasi-periodic initial data

Abstract

We study the Cauchy problem for the improved Boussinesq equation \[ u_{tt}-u_{xx}-u_{xxtt}-(u^2)_{xx}=0 \] on the real line with spatially quasi-periodic initial data. For a non-resonant frequency vector $\omega \in {\mathbb{R}}^{\nu }$, we prove local existence and uniqueness of classical spatially quasi-periodic solutions with the same frequency vector $\omega$ in two Fourier-side classes. First, for exponentially decaying initial Fourier coefficients, we obtain a spatially quasi-periodic solution whose Fourier coefficients remain exponentially decaying on an explicit time interval. Second, for initial Fourier coefficients $c(n)$ and $d(n)$ satisfying the polynomial decay $|c(n)|+|d(n)|\lesssim (1+|n|{)}^{-r},r>\nu +2,$ we prove that the corresponding spatially quasi-periodic solution preserves the same polynomial decay rate as the initial data. We also extend these results to the nonlinearity $u^p$ with integer $p\ge 3$.

33 Pages, 1 Figure