離散耦合非線性Schr?dinger格駐波解的存在性

趙 昕

(吉林農(nóng)業(yè)大學(xué) 信息技術(shù)學(xué)院,長春 130118)

在凝聚態(tài)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(tǒng):

(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(shè),晶格系統(tǒng)(1)可視為時(shí)變非線性Gross-Pitaevskii系統(tǒng)中二元物系統(tǒng)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(shù);?為普朗克常數(shù)除以2π;m為原子質(zhì)量;Vi為第i個(gè)超精細(xì)狀態(tài)的勢阱;ai≥0表示相關(guān)的軸頻率(i=1,2,3,4).系統(tǒng)(2)應(yīng)用于由兩種不同的超精細(xì)“玻色-愛因斯坦”凝聚態(tài)[1]組成的二元混合物中,目前關(guān)于系統(tǒng)(2)的駐波研究已有很多結(jié)果[2-6],關(guān)于離散非線性系統(tǒng)的局部化駐波解的存在性研究也取得了一定的進(jìn)展.如Aubry等[7-9]通過駐波的存在提出了反延續(xù)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(jié)構(gòu)的數(shù)學(xué)技巧有動(dòng)力系統(tǒng)技巧、變分方法和拓?fù)浼夹g(shù)等,如同倫方法[10]、山路引理[11]、Nehari流形[12]和Krasnoselskii不動(dòng)點(diǎn)定理[3].本文用Nehari流形方法研究晶格系統(tǒng)(1)駐波解的存在性.

設(shè)

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(jià)方程組:

(4)

考慮

(-Aφk,ψk)=(B*Bφk,ψk)=(Bφk,Bψk),

進(jìn)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(xiàn)[8-9,12]的結(jié)果,可得下列3個(gè)引理.

引理1假設(shè)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(shù)

(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(shù)E(φ,ψ)∈1(l2×l2,).

(8)

從而定理1的證明可歸結(jié)為如下兩個(gè)命題.

命題1序列{φn}和{ψn}分別在Hilbert空間S1和S2內(nèi)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(shù)系統(tǒng)(4)的一個(gè)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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