The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers (Dinvay, 2019; Dinvay et al., 2019) the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler–Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.

正在考虑对特定的双向Whitham系统建模地表水波的行波问题。该系统首先出现在Dinvay等人。（2019），在数值上显示它是稳定的，并且是不可压缩的Euler方程的良好近似。在随后的论文中（Dinvay，2019; Dinvay et al。，2019），研究了初值问题并证明了经典Sobolev空间中的适定性。在这里，我们证明了孤立波解的存在性，并提供了它们的渐近描述。我们的证明依赖于变分方法和浓度紧凑性论证。主要的困难源于以下事实：在考虑的Euler-Lagrange方程中，线性和非线性部分中都出现了一个正阶非局部算子。