具有PT对称性的周期Gross-Pitaevskii方程的非线性束缚态分支
Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with PT-symmetry
摘要：
本文考虑一维定常Gross-Pitaevskii方程，其复周期势满足PT（宇称-时间反转）对称性条件。在相当一般的势假设下，我们证明了具有复PT对称周期势的非自伴线性薛定谔算子谱中实区间端点的PT对称非线性束缚态的分岔。非线性束缚态近似的有效振幅方程，它具有立方非线性薛定谔方程的形式。此外，当复$\PT$对称势具有渐近小的虚部时，我们给出了复谱带出现的充分条件。
英文摘要：
The stationary Gross-Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the PT (parity-time reversal) symmetry. Under rather general assumptions on the potentials we prove bifurcations of PT-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrodinger operator with a complex PT-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrodinger equation. In addition we provide sufficient conditions for the appearance of complex spectral bands when the complex $\PT$-symmetric potential has an asymptotically small imaginary part.
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