大深度分层流体的界面波动是一种常见的物理现象，广泛存在于海洋、大气及工业流体等多种分层流体中。界面波动的生成、演化及相互作用是流体力学的重要研究方向，界面波之间的相互作用会产生更加复杂的流体环境，不仅对流体中的工程建设、军事活动等影响巨大，还与生态环境、气候变化和自然灾害等密切相关。为了深入研究大深度分层流体中的波–波相互作用问题，本文通过多尺度建模和数值模拟的方法，围绕大气中平面斜孤立波相互作用和重力毛细界面波的生成、演化及相互作用机制两类问题展开研究。

首先，建立了描述低层大气内波和重力毛细界面波的简化方程。使用Ablowitz-Fokas-Musslimani (AFM) 方法和Dirichlet-Neumann (DtN) 算子展开法求解复杂的三维非线性欧拉方程组，使用分界面上的波高和速度势重新表述原方程组，使其消除对垂直坐标的依赖。对维度降低的方程组进行泰勒展开和算子展开，最终根据需要保留合理阶数而得到模型方程。使用这两种方法分别推导了描述大气内波的Benjamin-Ono-Benney-Luke (BOBL) 方程和描述重力毛细界面波的Benjamin-Benney-Luke (BBL) 方程。

然后，基于BOBL 方程对大气中V 型平面斜孤立波的相互作用开展了数值研究。根据Benjamin-Ono 方程的解析解，推导了BOBL 方程的初值。基于此研究了V 型初值的开口方向、倾角、振幅、演变范围及地形对两列平面斜孤立波相互作用的影响。开口方向不同的四种V 型波的演化结果表明，交点前面的波对交点处的波形起加强作用，而交点后的波对交点波形起减弱作用；另外开口向右的V 型（简称V 型）波会产生非常复杂且与倾角密切相关的波形，于是进一步对这种V 型波进行深入研究。首先分析了倾角对V 型波相互作用的影响，数值结果表明演化结果大致可以分为两类，大倾角对应的规则反射和中等倾角及小倾角对应的马赫反射。与规则反射不同，马赫反射的主要特征为马赫杆的产生和演化。然后分析了振幅对V 型波相互作用的影响，当构成V 型波的两个分支振幅相等时，改变振幅后波形没有明显改变。但是当两个分支振幅不相等时，马赫杆两端速度不相等最终会产生倾斜的马赫杆。进一步考虑了V 型波在较大范围内传播的情况，此时色散效应占主导地位，不会产生相干波结构，最终波形会因色散而弯曲、衰减。最后，研究了外部环境对V 型波相互作用的影响。当系统考虑高度与振幅同阶的上凸型、下凹型和混合型三种地形时，波形以及马赫杆的演化特性均与平坦地形结果没有明显区别，说明小地形对相互作用的影响十分有限。

第三，基于BOBL 方程进一步分析了大气中更为复杂的X 型和Y 型平面斜孤立波的相互作用。倾角和振幅均在X 型波的相互作用中发挥重要作用，由于倾角的变化，X 型波演化过程也可以分为规则反射和马赫反射。在规则反射中，除了尾波和微小相移外波形基本不变，然而在马赫反射中，在产生马赫杆的同时交点后的两列波振幅不断减小，减小程度与倾角有关。通过振幅不相等的X 型波的计算结果可知，不相等的振幅会产生倾斜的马赫杆和更加复杂的波形。倾角对Y 型平面斜孤立波相互作用结果表明，Y 型结构会使发生马赫杆的倾角范围变大。最后讨论了周期地形对X 和Y 型相互作用的影响，尽管对波形的调制效果有限，但是周期地形对马赫杆的振幅、长度及速度的影响十分明显。

最后，基于BBL 方程研究了重力毛细界面波的生成、演化和相互作用的机理。针对非线性偏微分方程的刚性问题，推广了用于求解三维水波问题的指数差分格式。基于此开展了大量数值计算，得到了BBL 方程的完全局域的lump 解和分岔曲线。同时分析了平面孤立波的稳定性，当受到长波扰动时平面孤立波演化为一组lump，进一步证明了BBL 方程仅存在lump 解。为了研究lump 解的特性，开展了迎面碰撞、追赶碰撞和斜向碰撞，计算结果表明迎面碰撞为弹性碰撞，而追赶碰撞和斜向碰撞伴随明显的能量转移、相移等非线性现象，是非弹性碰撞。 

此外，提出了一种新的产生lump 解的方法，当流体以接近临界波速的速度经过局部地形时，会周期性的产生lump，但是当流速小于或者大于临界波速时只存在稳态的局部凹陷或V 型波。在此基础上，计算了匀加速流体经过局部地形的界面波形，由于流动的复杂性，此时会出现复合界面波形。

The interface waves of deep stratified fluids represent a prevalent physical phenomenon that widely exists across multiple stratified fluid systems, encompassing marine, atmospheric, and industrial fluids. The generation, evolution, and interaction of interface waves constitute a critical research area in fluid mechanics, as their mutual interactions can lead to highly complex fluid environments that have significant implications for various engineering and military operations in fluid systems, as well as close connections with ecological environments, climate change, and natural disasters.
In order to investigate in depth the wave-wave interaction phenomena in deep stratified fluids, this paper conducts research on two categories of problems based on multi-scale method and numerical simulations: the interaction between plane oblique solitary waves in the atmosphere and the generation, evolution, and interaction mechanisms of gravitycapillary
interfacial waves.

Firstly, the simplified equations for internal waves in the lower atmosphere and gravity-capillary interfacial waves are established. To solve the complex three dimensional nonlinear Euler equation system, the Ablowitz-Fokas-Musslimani (AFM) method and Dirichlet-Neumann (DtN) operator expansion method are used to reformulate the original equation system using the wave amplitude and velocity potential on the interface,
eliminating the dependence on the vertical coordinate. Furthermore, the Taylor and operator expansions are applied to obtain the Benjamin-Ono-Benney-Luke (BOBL) equation for atmospheric internal waves and the Benjamin-Benney-Luke (BBL) equation for gravity-capillary interfacial waves in accordance with the required order of accuracy.

Secondly, numerical investigations are implemented on the interaction of V-shaped plane oblique solitary waves in the atmosphere based on the BOBL equation. According to the analytical solution of the Benjamin-Ono equation, the initial value of the BOBL equation is derived, and the effects of the opening direction, inclination angle, amplitude, evolution range, and topography on the interactions of V-shaped waves are studied. The evolutions of four types of V-shaped waves with different opening directions reveal that the wave in front of the intersection enhances the wave shape at the intersection, while the wave behind it weakens it. Furthermore, it is evident that the interactions of V-shaped waves with a rightward opening (referred to as V-shaped) generate more complex wave patterns that are closely linked to the inclination angle, thereby necessitating additional investigation of this specific type of wave. The effects of the inclination angle on the interaction of V-shaped waves are conducted, which reveal that the results can be roughly divided into two categories: regular reflection for large inclination angles,and Mach reflection for moderate and small inclination angles, characterized by the production and evolution of a Mach stem. The impacts of amplitude on the interaction of V-shaped waves are investigated, revealing that the wave pattern remained unchanged when the two branches of the V-shaped waves have equal amplitudes, however, when the amplitudes of the two branches are unequal, a tilted Mach stem is observed due to the unequal velocities at the ends of the Mach stem. In the case of V-shaped waves propagating over a larger range, the dominant dispersion effect results in the absence of coherent wave structures, causing the wave pattern to both bend and decay. In the end, the influences of external environments on wave-wave interactions are explored, specifically three types of terrain with heights and amplitudes comparable in magnitude—upward convex, downward concave, and mixed terrain—on the interaction of V-shaped waves. Obviously, the evolution features of the wave pattern and Mach stem are similar to those with flat bottom, implying that the influence of small terrain on the interaction is limited, and the complexity of terrain has a negligible effect on the interaction of V-shaped waves. 

Thirdly, based on BOBL equation, the more complex interactions of X- and Y-shaped oblique solitary waves in the atmosphere are further discussed. Both inclination angles and amplitudes play important roles in the interaction of X-shaped waves. The evolution of X-shaped waves can also be divided into regular reflection and Mach reflection due to the variation of inclination angle. In regular reflection, the waveform is basically unchanged except for the tail wave and small phase shift. However, in Mach reflection, the amplitude of the two waves after the intersection of the Mach stem is reduced continuously, and the degree of reduction is related to the inclination angle. According to the calculation results of X-shaped interactions with different amplitudes, there exist inclined Mach stem and more complex waveform. The results of the interaction between inclination angle and Y-shaped plane oblique solitary wave show that the Y-shaped structure will make the range of inclination angle of Mach rod larger. Finally, the effect of periodic terrain on X and Y interaction is discussed. Although the modulation effect on waveform is limited, the effect of periodic terrain on the amplitude, length and velocity of Mach stem is very obvious.

Finally, the mechanism of generation, evolution, and interactions of gravity capillary interface waves are studied based on the BBL equation. To address the stiffness problem of the nonlinear partial differential equation, the exponential difference scheme is generalized to solve three-dimensional water wave problems. Based on this numerical method, extensive numerical calculations are carried out, and the local lump solution and bifurcation curve of the BBL equation are obtained. Then the stability of plane solitary waves is also analyzed. When subjected to long wave perturbations, plane solitary waves evolved into a group of lumps, further proving that the BBL equation onlyhas lump solutions. In order to study the characteristics of the lump solutions, head-on collisions, overtaking collisions, and oblique collisions are performed. The calculation
results showed that head-on collisions are elastic collisions, while overtaking collisions and oblique collisions are accompanied by obvious nonlinear phenomena such as energy transfer and phase shift, and are non-elastic collisions. In addition, a new method for producing lump solutions is proposed. When the fluid flowed over local terrain at a speed close to the critical wave speed, lumps are periodically generated. However, when the flow rate is smaller or larger than the critical wave speed, only a steady-state local depression or V-shaped wave existed. Based on this, the interface waveforms of uniformly accelerated fluids passing through local terrain are calculated. Due to the complexity of the flow, composite interface waveforms appeared at this time.