Title: Quasi-Gramian Solution of a Noncommutative Extension of the Higher-Order Nonlinear Schrödinger Equation Show Chinese title

Title: 非交换扩展的高阶非线性薛定谔方程的拟Gram矩阵解

Abstract: The nonlinear Schr{\"o}odinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation (HNLS). We treat real or complex-valued functions, such as g1 = g1(x, t) and g2 = g2(x, t), as non-commutative, and employ the Lax pair associated with the evolution equation as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation (DT). Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns in the context of modulational instability. Show Chinese abstract

Abstract: 非线性薛定谔（NLS）方程，包含高阶色散项，在各种物理现象的理论分析中被广泛使用。 在本研究中，我们探讨了高阶NLS方程（HNLS）的非交换扩展。 我们将实值或复值函数，如g1 = g1(x, t)和g2 = g2(x, t)视为非交换的，并像在交换情况下一样，采用与演化方程相关的Lax对。 通过采用二元达布变换（DT），我们推导出该系统的准Gramian解。 此外，该解可用于研究平面波的稳定性，并理解在调制不稳定性背景下的周期性图案的生成。