随着科学技术的发展与进步，等离子体已经成为物理学中最重要的研究对象之一。Alfvén波是等离子体中一种沿着磁场方向传播的波，它在高能物理、天体力学中具有广泛的应用。在数学上，Alfvén波的动力学行为可以由一组被称为磁流体力学（MHD）方程组的非线性偏微分方程系统来描述。然而，由于MHD方程组很复杂，目前很少有合适的数学理论能严格证实所观察到的物理现象，特别比如Alfvén波散射问题的刚性现象，即如果在无穷远处的观察者没有探测到任何波，那么说明等离子体本来就没有产生Alfvén波。该现象具有深刻的物理背景，这启发我们提出研究Alfvén波在无穷远处散射行为的新方法。在对非线性Alfvén波的研究中，我们注意到它的动力学行为类似于一维波，并且在强背景磁场下的MHD方程组蕴含了具有光结构的非线性项。因此，为了更好地描述Alfvén波在无穷远处的行为，本文首先对具有光形式非线性项的一维半线性波动方程模型开展研究，针对一维半线性波的散射问题给出在无穷远处的刚性。之后，本文回到三维理想的不可压 MHD 方程组。基于Alfvén波散射场的定义，我们将无穷远处散射场逐点为零的条件转化为解在有限大时间下的L2-小性条件，进而根据能量估计推出非线性Alfvén波在无穷远处的一组刚性定理，即如果Alfvén波在无穷远处的散射场为零，那么Alfvén波本身一定是消失的。对比前人关于线性波动方程的刚性研究，本文提出的刚性结果体现了以下两方面亮点：一是本文建立在非线性背景下，充分考虑了MHD方程组的拟线性性质；二是本文只需要散射场本身为零的条件，这与刚性的物理直观一致。关于方法上的创新，本文先通过几何方法构思了散射无穷远以及Alfvén波的散射场，再结合波动方程的观点与加权能量估计的方法，以及提出在所加权重中引入追踪Alfvén波中心的位置参数，进而严格建立了Alfvén波的散射场与等离子体的刚性之间的关系。由此展望，本文为描述波动方程解的长时间行为提供了新视角，为研究MHD方程组的散射理论提供了新技术，也为构建波在无穷远处的刚性提供了新思路。

With the development of science and technology, plasmas have become some of the most important and intensively studied objects in physics. The Alfvén waves, propagating along the direction of magnetic field, are fundamental wave phenomena in magnetized plasmas and have a wide range of applications in high energy physics and astrophysics. Mathematically, the dynamics of Alfvén waves are governed by a system of nonlinear partial differential equations called the magnetohydrodynamics (MHD) equations. However, due to the complexity of the MHD equations, there are few suitable and rigorous mathematical explanation for physical phenomena, especially the rigidity phenomenon about the scattering problem of Alfvén waves. In other words, if no waves are detected by the far-away observers, then there are no Alfvén waves at all emanating from the plasma. This phenomenon has a profound physical background, which motivates us to propose new methods to study the scattering behavior of Alfvén waves.We note that the dynamic behavior of nonlinear Alfvén waves is similar to that of one-dimensional waves, and the MHD equations in strong magnetic backgrounds contain nonlinear terms with null structure. Therefore, in order to describe the scattering behavior of Alfvén waves more naturally, we first show the rigidity from infinity for the model of one-dimensional semilinear wave equations verifying the null conditions. Later on, this dissertation returns to the three-dimensional ideal incompressible MHD equations. By introducing the scattering fields of Alfvén waves, we translate the point-wise vanishing properties of the scattering fields at infinities to the L2-smallness conditions for solutions at a large finite time. According to these preparations, we then apply a family of weighted energy estimates to derive a couple of rigidity from infinity theorems that the Alfvén waves must vanish if their scattering fields vanish at infinities. Compared with the rigidity of linear wave equations in previous works, the rigidity results provided in this dissertation reflect the following two highlights. One is that reasoning herein is based on nonlinear setting and hence has thoroughly examined the quasilinear nature of the MHD equations. The other is that our assumption on scattering fields is more consistent with the physical intuition of rigidity. This is because we merely require that the scattering fields themselves are equal to zero at infinities, in contrast with requiring higher order derivatives of scattering fields to vanish. Regarding innovations in the strategy, we first introduce the infinities and the scattering fields of Alfvén waves through geometric methods, then synthesize the idea of wave equations and the weighted energy estimates, and also architect a position parameter in weights to trace the centers of Alfvén waves. As such, we establish the relationship between the scattering fields of Alfvén waves and the rigidity of plasmas. On?this?basis, this dissertation opens up a new perspective to describe the long time behavior of solutions to wave equations, creates an original way to study the scattering theory of MHD equations, and also develops a novel approach to constructing the rigidity from infinity for waves.