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PDF(154 KB)
PDF(154 KB)
扰动KdV方程解的先验估计
Priori estimates for the solutions of perturbed Korteweg-de Vries Equation
对于带微扰的KdV方程ut+6uux+uxx=εR(u)，(ε＞0)，在初值u0(x)∈C∞(-∞,+∞)，当|x|→∞时指数衰减的条件下，分别构造出带两种不同扰动项的KdV方程的扰动孤立波解满足的能量关系式，并运用能量分析方法对扰动的孤立波解进行先验估计，得到如下结论：(1)R(u)=δ(εt)u, δ(s)∈C［0,+∞)，δ(0)=0，时，解在-∞＜x＜+∞，0≤εt≤T内一致有界；(2)R(u)=-Δ(εt)uxxx，Δ(0)=0，Δ(s)∈C1［0,+∞), 解在-∞＜x＜+∞，0≤εt≤T，0≤ε≤ε1内一致有界。
Energy equalities are constructed for the perturbed solitary wave solutions corresponding to two kinds of perturbations for the perturbed KdV equation
ut+6uux+uxx=εR(u)，(ε＞0), under the condition that the initial data u0(x)∈C∞(-∞,+∞), decay exponentially as |x|→∞. Priori estimates of the bound of the solutions are obtained via the method of energy analysis: (1) if R(u)=δ(εt)u, δ(s)∈C［0,+∞) and δ(0)=0, the solutions are uniformly bounded in the region -∞＜x＜+∞，0≤εt≤T; (2) in the case of R(u)=-Δ(εt)uxxx, Δ(0)=0, Δ(s)∈C1［0,+∞), the solutions are uniformly bounded in the region -∞＜x＜+∞, 0≤εt≤T, 0≤ε≤ε1 for some positive small ε1.
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