Acta mathematica scientia, Series B >
PERTURBED PERIODIC SOLUTION FOR BOUSSINESQ EQUATION
Received date: 2008-12-24
Online published: 2009-05-20
Supported by
This work is supported by National Natural Science Foundation of China (10871199)
We consider the solution of the good Boussinesq equation
Utt − Uxx + Uxxxx = (U2)xx, −∞ < x < ∞, t ≥ 0,
with periodic initial value
U(x, 0) = ∈(μ + Φ(x)), Ut (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.
JIANG Xin-Hua , WANG Zhen . PERTURBED PERIODIC SOLUTION FOR BOUSSINESQ EQUATION[J]. Acta mathematica scientia, Series B, 2009 , 29(3) : 705 -722 . DOI: 10.1016/S0252-9602(09)60066-7
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