标量和矢量非线性Schr？dinger型方程局部模的全局搜索
Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations
摘要：
我们提出了一种新的方法搜索共存类的排斥（散焦）标量或矢量非线性薛定谔型方程承认的本地化模式。该方法是基于观察到，相应的固定系统的通用解决方案有奇点在有限点的实轴上。我们首先建立条件的初始数据相关联的柯西问题，保证形成一个奇点。利用这些充分条件，我们确定有界的，非奇异的，解决方案-然后根据他们的渐近行为进行分类。为了确定有界的解决方案，一个适当选择的空间的初始数据进行数字扫描。由于渐近或对称的考虑，我们可以将自己限制在一维或二维空间。对于每一组初始条件，我们计算距离$X^{\pm}$最近的前向和后向奇点;大$X^+$或$X^-$表示接近有界解。我们说明了我们的方法与Gross-Pitaevskii方程与$\PT$ -对称的复势，一个系统的耦合Gross-Pitaevskii方程与实际的潜力，和Lugiato-Lefever方程正常色散。
英文摘要：
We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schrödinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions --- and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances $X^{\pm}$ to the nearest forward and backward singularities; large $X^+$ or $X^-$ indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a $\PT$-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion.
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