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

TL;DR

This paper establishes local existence and uniqueness of quasi-periodic solutions for an improved Boussinesq equation with specific decay properties of initial data, extending results to nonlinearities of higher order.

Contribution

It provides the first rigorous proof of local well-posedness for the improved Boussinesq equation with quasi-periodic initial data, including polynomial decay cases and nonlinear extensions.

Findings

Existence of solutions with exponentially decaying initial Fourier coefficients.

Preservation of polynomial decay rates in solutions over time.

Extension of results to nonlinearities u^p with p ≥ 3.

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 $p \geq 3$.