離散耦合非線性Schr?dinger格駐波解的存在性

趙 昕

(吉林農業大學 信息技術學院,長春 130118)

在凝聚態物理中,非線性Schr?dinger格用于描述晶格的演變.考慮如下耦合離散Schr?dinger系統:

(1)

其中:ε>0;k∈;是有界位勢(i=1,2,3);A 是離散Laplace算子,(Au)k=uk+1+uk-1-2uk.

通過尺度變換和一些簡單的假設,晶格系統(1)可視為時變非線性Gross-Pitaevskii系統中二元物系統的空間離散化形式:

(2)

其中:u(x,t)和v(x,t)為凝聚物波函數;?為普朗克常數除以2π;m為原子質量;Vi為第i個超精細狀態的勢阱;ai≥0表示相關的軸頻率(i=1,2,3,4).系統(2)應用于由兩種不同的超精細“玻色-愛因斯坦”凝聚態[1]組成的二元混合物中,目前關于系統(2)的駐波研究已有很多結果[2-6],關于離散非線性系統的局部化駐波解的存在性研究也取得了一定的進展.如Aubry等[7-9]通過駐波的存在提出了反延續或反可積性的概念,深入探討了離散非線性Schr?dinger方程.研究該結構的數學技巧有動力系統技巧、變分方法和拓撲技術等,如同倫方法[10]、山路引理[11]、Nehari流形[12]和Krasnoselskii不動點定理[3].本文用Nehari流形方法研究晶格系統(1)駐波解的存在性.

設

uk(t)=exp(-iω1t)φk,vk(t)=exp(-iω2t)ψk,k∈,

(3)

將式(3)代入式(1),可得如下等價方程組:

(4)

考慮

(-Aφk,ψk)=(B*Bφk,ψk)=(Bφk,Bψk),

進而有

即A:l2→l2為有界算子且σ(-A)?[0,4].定義在l2中的自伴算子:

K1=-εA+V1-2V2, K2=-εA+V3-2V2.

(5)

(H2) 頻率ω1和ω2滿足

仿照文獻[8-9,12]的結果,可得下列3個引理.

引理1假設條件(H1)成立,令

則對于任意的p(2≤p≤∞),S1和S2緊嵌入到lp中,并且譜σ(K1)和σ(K2)分別是離散的.

定義能量函數

(6)

和Nehari流形

N∶={(φ,ψ)∈l2×l2: H1(φ,ψ)=H2(φ,ψ)},

(7)

其中:

注1由引理1可知,能量函數E(φ,ψ)∈1(l2×l2,).

(8)

從而定理1的證明可歸結為如下兩個命題.

命題1序列{φn}和{ψn}分別在Hilbert空間S1和S2內有界,并且存在φ*∈S1和ψ*∈S2,使得φn→φ*,ψn→ψ*∈l2.

命題2(φ*,ψ*)∈N且E(φ*,ψ*)=m,則 (φ*,ψ*)是代數系統(4)的一個非平凡弱解.

由引理1～引理3,再利用Nehari流形技巧,即可證明命題1和命題2,從而定理1成立.

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