关键词: Boussinesq equation, periodic solution, Bäcklund transformation, global existence, uniform asymptotic cxpansion

Abstract:

We consider the solution of the good Boussinesq equation
U_tt − U_xx + U_xxxx = (U²)_xx, −∞ < x < ∞, t ≥ 0,
with periodic initial value
U(x, 0) = ∈(μ + Φ(x)), U_t (x, 0) =∈ψ (x), −∞ < x < ∞,
where μ ≠ 0, Φ(x) and  ψ(x) are 2-periodic functions with 0-average value in [0, 2π], and ∈ is small. A two parameter Bäcklund 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 ", 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,  (x) = b cos kx, 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; ∈) and the N-th partial sum of the asymptotic series is bounded by ∈^N+1 multiplied by a constant depending on T and N, for all −∞ < x < ∞, 0 ≤ |∈|t ≤ T and 0 ≤ |∈| ≤ ∈0.

Key words: Boussinesq equation, periodic solution, Bäcklund transformation, global existence, uniform asymptotic cxpansion