Perturbed periodic solution for boussinesq equation

We consider the solution of the good Boussinesq equation with periodic initial value where μ ≠ 0, ϕ(x) and ψ(x) are 27π-periodic functions with 0-average value in [0, 2π], and ɛ is small. A two parameter Backhand transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ɛ, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form ϕ(x)=μ + a sin kx, ψ(x)= an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T > 0, the difference between the true solution u(x,t;ɛ) and the N-th partial sum of the asymptotic series is bounded by φN+1 multiplied by a constant depending on T and N, for all −∞ < x < ∞, 0 ⩽ |ɛ|t and 0⩽|ɛ| ⩽ ɛ0.