Attractors of Weakly Damped Forced KdV Equations with External Periodic Excitation

摘要

1　Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎｔｈｅｐａｓｔｄｅｃａｄｅｏｒｓｏ ,ｔｈｅｅｘｉｓｔｅｎｃｅｏｆｇｌｏｂａｌａｔ ｔｒａｃｔｏｒｓａｎｄｔｈｅｅｓｔｉｍａｔｅｓｏｆｔｈｅｉｒＨａｕｓｄｏｒｆｆａｎｄｆｒａｃｔａｌｄｉｍｅｎｓｉｏｎｓｈａｖｅｂｅｅｎｉｎｖｅｓｔｉｇａｔｅｄｆｏｒｎｕｍｅｒｏｕｓｄｉｓｓｉｐａ ｔｉｖｅｎｏｎｌｉｎｅａｒｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｈｉｃｈａｒｅａｕ ｔｏｎｏｍｏｕｓ[1～ 4 ].Ｔｈｅｎｏｎ ａｕｔｏｎｏｍｏｕｓｉｎｆｉｎｉｔｅ ｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓｗｅｒｅｌｅｓｓｕｎｄｅｒｓｔｏｏｄ .ＩｎｔｈｉｓｐａｐｅｒｗｅｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇＫｄＶｅｑｕａｔｉｏｎｓｗｉｔｈｅｘｔｅｒｎａｌｐｅｒｉｏｄｉｃｅｘ ｃｉｔａｔｉｏｎ :ｕｔ+ｕｕｘ+ｕｘｘｘ +γｕ=ｆ(ｔ) ,(1)ｕ(ｘ+Ｌ ,ｔ) =ｕ(ｘ ,ｔ) ,(2 )ｕ｜ｔ=τ =ｕ0 (ｘ) ∈Ｈ2 Ｌ,(3)ｗｈｅｒｅｆ∈Ｌ∞(0 ,Ｔ ,Ｈ2 Ｌ) ,ｆ(ｔ+Ｔ) =ｆ(ｔ) , ｔ∈Ｒ .Ｗｅｉｎｔｒｏｄｕｃｅｔｈｅｆａｍｉｌｙｏｆｏｐｅｒａｔｏｒｓ ...