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Asymptotic Analysis of a Perturbed Periodic Solution for the KdV Equation
Author(s) -
Jiang X. H.,
Wong R.
Publication year - 2006
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2005.00332.x
Subject(s) - korteweg–de vries equation , mathematics , bounded function , constant (computer programming) , mathematical analysis , image (mathematics) , series (stratigraphy) , asymptotic expansion , mathematical physics , energy (signal processing) , initial value problem , physics , nonlinear system , quantum mechanics , paleontology , statistics , computer science , biology , programming language , artificial intelligence
We consider the solution of the Korteweg–de Vries (KdV) equationwith periodic initial valuewhere C , A , k , μ, and β are constants. The solution is shown to be uniformly bounded for all small ɛ, and a formal expansion is constructed for the solution 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 v ( x , t ; ɛ) and the N th partial sum of the asymptotic series is bounded by ɛ N +1 multiplied by a constant depending on T and N , for all −∞ < x < ∞, 0 ≤ t ≤ T /ɛ , and 0 ≤ɛ≤ɛ 0 .