一维非线性薛定谔方程中支持表面带隙孤子基态的界面
Interfaces Supporting Surface Gap Soliton Ground States in the 1D Nonlinear Schroedinger Equation
摘要：
本文考虑了一维非线性薛定谔方程的$H^1$基态存在性的验证问题，其中$x\geq 0$和$x<0$分别为$$-u" +V(x)u -λu = Γ(x) |u|^{p-1}u \ {on} \R$$和$V(x) = V_1(x), Γ(x)=Γ_1(x)$，$V(x) = V_2(x), Γ(x)=Γ_2(x)$.这里$V_1,V_2,Γ_1,Γ_2$是周期性的，$λ<\minσ(-\tfrac{d^2}{dx^2}+V)$和$p>1$。文章[T。Dohnal，M. Plum和W. Reichel，“Surface Gap Soliton Ground States for the Nonlinear Schrödinger Equation，”\textit{Comm. Math. Phys.} \textbf{308}，511-542（2011）]在1D情况下提供了以积分不等式形式的存在性准则，该积分不等式涉及算子$-\tfrac{d^2}{dx^2}+V_{1,2}-λ$的线性势$V_{1},V_2$和Bloch波。我们在这里选择的类分段常数和分段线性电位$V_{1,2}$和检查这个标准的一组参数值。在分段常数的情况下，Bloch波的显式计算和在分段线性的情况下，验证外壳的Bloch波的数值计算。该准则中的积分通过区间算术进行评估，从而产生严格的存在性声明。支持基态的接口的示例被报告，包括对于所有周期性$Γ_ {1,2}$与$\esssup Γ_{1,2}>0$的基态存在。
英文摘要：
We consider the problem of verifying the existence of $H^1$ ground states of the 1D nonlinear Schrödinger equation for an interface of two periodic structures: $$-u" +V(x)u -λu = Γ(x) |u|^{p-1}u \ {on} \R$$ with $V(x) = V_1(x), Γ(x)=Γ_1(x)$ for $x\geq 0$ and $V(x) = V_2(x), Γ(x)=Γ_2(x)$ for $x<0$. Here $V_1,V_2,Γ_1,Γ_2$ are periodic, $λ<\minσ(-\tfrac{d^2}{dx^2}+V)$, and $p>1$. The article [T. Dohnal, M. Plum and W. Reichel, "Surface Gap Soliton Ground States for the Nonlinear Schrödinger Equation," \textit{Comm. Math. Phys.} \textbf{308}, 511-542 (2011)] provides in the 1D case an existence criterion in the form of an integral inequality involving the linear potentials $V_{1},V_2$ and the Bloch waves of the operators $-\tfrac{d^2}{dx^2}+V_{1,2}-λ$. We choose here the classes of piecewise constant and piecewise linear potentials $V_{1,2}$ and check this criterion for a set of parameter values. In the piecewise constant case the Bloch waves are calculated explicitly and in the piecewise linear case verified enclosures of the Bloch waves are computed numerically. The integrals in the criterion are evaluated via interval arithmetic so that rigorous existence statements are produced. Examples of interfaces supporting ground states are reported including such, for which ground state existence follows for all periodic $Γ_ {1,2}$ with $\esssup Γ_{1,2}>0$.
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