阻尼驱动KdV方程及其解长时间行为的有效方程
偏微分方程分析
2010-02-08 v1 数学物理
math.MP
摘要
对于阻尼驱动KdV方程 u ˙ − ν u x x + u x x x − 6 u u x = ν η ( t , x ) , x ∈ S 1 , ∫ u d x ≡ ∫ η d x ≡ 0 , \dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x), x\in S^1, \int u dx\equiv \int\eta dx\equiv0, u ˙ − ν u xx + u xxx − 6 u u x = ν η ( t , x ) , x ∈ S 1 , ∫ u d x ≡ ∫ η d x ≡ 0 , 0 < ν ≤ 1 0<\nu\le1 0 < ν ≤ 1 η \eta η ν → 0 \nu\to0 ν → 0 u ν ( t , x ) u^\nu(t,x) u ν ( t , x ) ( I 1 , I 2 , . . . ) (I_1,I_2,...) ( I 1 , I 2 , ... ) 0 ≤ τ : = ν t ≲ 1 0\le\tau:= \nu t \lesssim1 0 ≤ τ := ν t ≲ 1 I ν ( τ ) = ( I 1 ( u ν ( τ , ⋅ ) ) , I 2 ( u ν ( τ , ⋅ ) ) , . . . ) , I^\nu(\tau)=(I_1(u^\nu(\tau,\cdot)),I_2(u^\nu(\tau,\cdot)),...), I ν ( τ ) = ( I 1 ( u ν ( τ , ⋅ )) , I 2 ( u ν ( τ , ⋅ )) , ... ) , I 0 ( τ ) = ( I 1 0 , I 2 0 , . . . ) I^0(\tau)=(I^0_1,I^0_2,...) I 0 ( τ ) = ( I 1 0 , I 2 0 , ... ) j j j I j 0 I_j^0 I j 0 1 2 ( v j ( τ ) 2 + v − j ( τ ) 2 ) \frac12(v_j(\tau)^2+v_{-j}(\tau)^2) 2 1 ( v j ( τ ) 2 + v − j ( τ ) 2 ) v ( τ ) = ( v 1 ( τ ) , v − 1 ( τ ) , v 2 ( τ ) , . . . ) v(\tau)=(v_1(\tau), v_{-1}(\tau),v_2(\tau),...) v ( τ ) = ( v 1 ( τ ) , v − 1 ( τ ) , v 2 ( τ ) , ... )
引用
@article{arxiv.1002.1294,
title = {Damped-driven KdV and effective equation for long-time behaviour of its solutions},
author = {Sergei B. Kuksin},
journal= {arXiv preprint arXiv:1002.1294},
year = {2010}
}