基于守恒量诊断的非线性薛定谔方程孤子传播与谱方法研究

Spectral Analysis of Soliton Propagation in the Nonlinear Schrödinger Equation Based on Conservation Diagnostics

摘要: 非线性薛定谔方程是数学物理中描述色散、非线性自聚焦和孤子传播的重要模型，如何在有限计算区域内稳定再现孤子形态并保持基本守恒量，是开展非线性波数值研究的关键问题。本文以一维聚焦型三次非线性薛定谔方程为对象，构造解析亮孤子解作为可验证基准，采用傅里叶谱离散处理空间二阶导数，并用Strang分步格式分离线性色散算子与非线性相位算子；同时引入质量、哈密顿量、相对L₂误差和末时刻振幅相关系数作为守恒量诊断与精度评价指标。数值结果表明，在周期边界与足够宽计算区间下，孤子在传播过程中保持明显的局域结构，时空密度图未出现可见弥散破坏；质量最大相对误差约为9.137e−14，哈密顿量最大相对误差约为1.637e−11，末时刻相对L₂误差约为6.814e−06，振幅散点基本贴近理想对角线。进一步的时间步长和空间网格收敛性实验表明，该方法在时间方向呈现接近二阶的收敛特征，空间方向表现出谱方法对光滑孤子解的快速收敛优势；双孤子碰撞算例也表明，所提出的守恒量诊断框架能够用于更复杂的非线性相互作用过程。基于守恒量诊断的傅里叶谱分步方法能够有效刻画非线性薛定谔孤子的长期传播特征，可为非线性波、量子流体和光学孤子等数学物理问题提供可复现的计算框架。

Abstract: The nonlinear Schrödinger equation is a fundamental mathematical-physics model for dispersion, nonlinear self-focusing and soliton propagation. A stable numerical description of soliton motion requires not only small pointwise errors but also reliable conservation diagnostics. Taking the one-dimensional focusing cubic nonlinear Schrödinger equation as the model problem, this study constructs a bright-soliton benchmark solution, discretizes the spatial derivative by a Fourier spectral method, and advances the solution by a Strang split-step scheme that separates the linear dispersive operator from the nonlinear phase operator. The mass, Hamiltonian, relative L₂ error and final-time amplitude correlation coefficient are used as diagnostic indicators. The numerical density map shows that the localized soliton structure is preserved during propagation. The maximum relative mass error is about 9.137e−14, the maximum relative Hamiltonian error is about 1.637e−11, and the final-time relative L₂ error is about 6.814e−06. The final-time amplitude scatter is almost aligned with the ideal diagonal. Further convergence tests with respect to the time step and spatial grid resolution show that the method exhibits nearly second-order accuracy in time and rapid spectral convergence for the smooth soliton solution. In addition, a two-soliton collision test demonstrates that the proposed conservation-diagnostic framework can also be extended to more complex nonlinear interaction scenarios. The conservation-diagnostic Fourier split-step framework provides a reproducible and physically interpretable tool for nonlinear wave problems in mathematical physics.