AI 中文总结
本文为Davey-Stewartson系统建立Whitham调制理论，分析其四种变体周期行波解的稳定性，发现仅散焦DSII系统线性稳定，其余均不稳定，并通过数值模拟验证了理论。
AI 中文摘要
Davey-Stewartson（DS）系统是二维空间中弱非线性波包的典型模型，该波包与其产生的缓变平均流共振耦合，因此它支配着不同场景下的调制波列，例如有限深度水波、二次光学介质和磁性薄膜。在此，我们为DS系统发展了Whitham调制理论，并利用该理论研究其周期行波解的稳定性。具体而言：1. 我们对DS系统所有四种变体的周期解进行了详细的参数依赖性研究，包括其谐波极限和孤子极限。2. 我们提出了DS系统的Whitham调制理论的公式化表述，并为DS系统的所有四种变体写下了由此产生的Whitham调制方程。3. 我们利用所得的调制方程来研究DS系统所有四种变体的周期行波解的稳定性。4. 我们通过将理论预测与DS系统的直接线性化以及DS系统的直接数值模拟进行比较来验证理论预测，显示出极好的一致性。结果表明，散焦DSII系统的周期解是线性稳定的，而DS系统所有其他变体的周期解是不稳定的。这些计算还揭示了许多结构特征（非演化调制方程、对允许初始数据的约束，以及行波解中不存在但调制系统不可或缺的辅助平均场常数），这些特征预计会在任何具有平均场耦合的二维包络方程中重现。
英文摘要
The Davey-Stewartson (DS) system is a canonical model for a weakly nonlinear wave packet in two spatial dimensions that is resonantly coupled to the slowly varying mean flow it generates, and as such it governs modulated wave trains in settings as different as finite-depth water waves, quadratic optical media and magnetic films. Here we develop the Whitham modulation theory for the DS system and use it to study the stability of its periodic traveling wave solutions. Specifically: 1. We perform a detailed study of the parametric dependence of the periodic solutions of all four variants of the DS system, including their harmonic and soliton limits. 2. We present the formulation of Whitham modulation theory for the DS system and write down the resulting Whitham modulation equations for all four variants of the DS system. 3. We use the resulting modulation equations to study the stability of the periodic traveling wave solutions of all four variants of the DS system. 4. We validate the theoretical predictions by comparing them with a direct linearization of the DS system as well as with the direct numerical simulations of the DS system, showing excellent agreement. The results indicate that the periodic solutions of the defocusing DSII system are linearly stable, whereas those of all other variants of the DS system are unstable. The calculations also evidence many structural features (non-evolutionary modulation equations, constraints on the admissible initial data, and auxiliary mean-field constants absent from the traveling wave solutions yet indispensable to the modulation system) are expected to recur in any two-dimensional envelope equation with mean-field coupling.
Comments53 pages, 4 figures
