This manuscript describes the derivation of systems of equations for weakly nonlinear gravity waves in shallow water in the presence of constant vorticity. The derivation is based on a multi-layer generalization of the traditional columnar Ansatz. A perturbative development in a nonlinear parameter and a dispersive parameter allow us to obtain sets of equations, for the horizontal fluid velocity and the free surface, able to describe propagation of weakly nonlinear and dispersive surface waves moving in water with some prescribed initial constant vorticity. We have shown that vorticity plays a central role on the dispersive properties of the system. When it is weak, it acts as a correction in linear and nonlinear dispersive terms. When stronger, it can also influence the nondispersive behavior of the system. Explicit steady solutions of the system corresponding to zero, weak, normal or strong vorticity are obtained. They correspond to solitary waves. Evolution of the soliton celerity, amplitude and width for these four cases are discussed.

该手稿描述了在存在恒定涡度的情况下浅水中的弱非线性重力波方程组的推导。该推导基于传统柱状Ansatz的多层概括。非线性参数和弥散参数的摄动发展使我们能够获得水平流体速度和自由表面的方程组，能够描述具有一定初始初始涡度的，在水中运动的弱非线性和弥散表面波的传播。我们已经表明，涡度在系统的色散特性中起着核心作用。当它很弱时，它可以作为线性和非线性色散项的校正。如果强度更高，它也会影响系统的非分散行为。获得了对应于零，弱，正或强涡度的系统的显式稳定解。它们对应于孤立波。讨论了这四种情况下孤子速度，幅度和宽度的演变。