We present a rigorous modulational stability theory for periodic traveling wave solutions to equations of nonlinear Schrödinger type. For Hamiltonian dispersive equations with a non-singular symplectic form and d conserved quantities (in addition to the Hamiltonian), one expects that generically \({{\mathcal {L}}}\), the linearization around a periodic traveling wave, will have a particular Jordan structure. The kernel \(\ker ({\mathcal L})\) and the first generalized kernel \(\ker ({\mathcal L}^2)/\ker ({{\mathcal {L}}})\) are expected to be d dimensional, with no higher generalized kernels. The breakup of this Jordan block under perturbations arising from a change in boundary conditions dictates the modulational stability or instability of the underlying periodic traveling wave. This general picture is worked out in detail for equations of nonlinear Schrödinger (NLS) type. We give explicit genericity conditions that guarantee that the Jordan form is the generic one: these take the form of non-vanishing determinants of certain matrices whose entries can be expressed in terms of a finite number of moments of the traveling wave solution. Assuming that these genericity conditions are met we give a normal form for the small eigenvalues that result from the break-up of the generalized kernel, in the form of the eigenvalues of a quadratic matrix pencil. We compare these results to direct numerical simulation for the cubic and quintic focusing and defocusing NLS equations subject to both longitudinal and transverse perturbations. The stability of traveling waves of the cubic NLS subject to longitudinal perturbations has been previously studied using the integrability and our results agree with those in the literature. All of the remaining cases are new.

对于非线性Schrödinger型方程的周期行波解，我们提出了严格的调制稳定性理论。对于具有非奇异辛形式和d守恒量（除了哈密顿量）的哈密​​顿色散方程，人们期望一般\（{{\ mathcal {L}}} \）（围绕周期行波的线性化）将具有特定的乔丹结构。内核\（\ KER（{\ mathcal L}）\）和所述第一广义内核\（\ KER（{\ mathcal L} ^ 2）/ \ KER（{{\ mathcal {L}}}）\）是预计d维度，没有更高的广义内核。在边界条件变化引起的扰动作用下，该乔丹块的破裂指示了基础周期性行波的调制稳定性或不稳定性。对于非线性Schrödinger（NLS）类型的方程，将详细了解此总体情况。我们给出明确的通用性条件，以保证Jordan形式是通用性：这些条件采用某些矩阵的不消失的行列式形式，其项可以用行波解的有限矩表示。假设满足了这些通用性条件，我们以平方矩阵铅笔的特征值的形式给出了归因于广义核分解的小特征值的范式。我们将这些结果进行比较，以对受到纵向和横向扰动的三次和五次聚焦和散焦NLS方程进行直接数值模拟。以前已经使用可积性研究了立方NLS受到纵向扰动的行波的稳定性，我们的结果与文献中的结果相符。其余所有案例都是新的。