具常边界条件的多分量非线性薛定谔方程
On the Multi-Component Nonlinear Schrödinger Equation with Constant Boundary Conditions
摘要：
关于C.I=Sp（2 p）/U（p）和D.III=SO（2 p）/U（p）型对称空间的多分量非线性Schrodinger方程，具有非零边界条件，可用逆散射方法（ISM）求解.作为Lax算子L，我们使用广义Zakharov-Shabat算子。我们表明，ISM L是一个非线性模拟的傅里叶变换方法。作为通常的傅里叶指数函数的适当推广，我们使用所谓的平方解，它是根据L的基本解析解（FAS）和李代数的Cartan-Weyl基构造的，与对称空间相关。我们推导出的平方解决方案，原来提供谱分解的递归（生成）运营商，一个自然的推广ID/DX的情况下，非线性发展方程的完整性关系。
英文摘要：
The multi-component nonlinear Schrodinger equation related to C.I=Sp(2p)/U(p) and D.III=SO(2p)/U(p)-type symmetric spaces with non-vanishing boundary conditions is solvable with the inverse scattering method (ISM). As Lax operator L we use the generalized Zakharov-Shabat operator. We show that the ISM for L is a nonlinear analog of the Fourier-transform method. As appropriate generalizations of the usual Fourier-exponential functions we use the so-called squared solutions, which are constructed in terms of the fundamental analytic solutions (FAS) of L and the Cartan-Weyl basis of the Lie algebra, relevant to the symmetric space. We derive the completeness relation for the squared solutions which turns out to provide spectral decomposition of the recursion (generating) operators, a natural generalizations of id/dx in the case of nonlinear evolution equations.
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