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The numerical study of the dynamics of two-dimensional capillary-gravity solitary waves on a linear shear current is presented in this paper. The numerical method is based on the time-dependent conformal mapping. The stability of different kinds of solitary waves is considered. Both depression wave and large amplitude elevation wave are found to be stable, while small amplitude elevation wave is unstable to the small perturbation, and it finally evolves to be a depression wave with tails, which is similar to the irrotational capillary-gravity waves.

A capillary-gravity wave on a fluid surface is influenced by both the effects of surface tension and gravity. The wavelength of capillary-gravity waves in water is typically less than a few centimeters; therefore, they are often referred to as ripples. On the open ocean, much larger surface water waves may result from coalescence of smaller wind-caused capillary-gravity waves. In the past several decades, the irrotational capillary-gravity waves have been studied by many scientists. In the beginning stage of the wind-driven wave, shear layer is often generated by the wind where the capillary-gravity waves often run on. So it is more realistic to consider the whole system in the presence of vorticity, and the simplest way is to incorporate a linear background shear in the potential flow.

Steady gravity waves with constant vorticity have been considered numerically by applied mathematicians in 1980s and 1990s. Many new profiles due to the shear effect have been found (see [

In the numerical aspects of two-dimensional surface water waves, the boundary integral method is proven to be very accurate and efficient for finding permanent wave profiles. For the unsteady computation, a new method based on time-dependent conformal mapping was derived by Dyachenko et al. [

In [

We consider 2D surface water waves and choose Cartesian coordinates so that the

Following the method presented in [

Due to the small length scale of the capillary-gravity waves (

Dispersion relations for different vorticity strengths

To study the dynamics of the capillary-gravity solitary waves with linear background shear, it is the first step to seek traveling wave solution to the Euler equations (

The nonlinear integral-differential equation (

(a) Amplitude versus speed bifurcation. There are two branches for each

Wave profiles with the same translating speed but different vorticity. Depression solitary waves at

For irrotational capillary-gravity solitary waves, the stability analysis has been carried out by Calvo and Akylas (2002), and later by Milewski et al. [

Typical profiles of two branches capillary-gravity solitary waves. Depression solitary wave (a) with

(a) Unstable elevation solitary wave with

Numerical experiments for the dynamics of the capillary-gravity solitary waves on a linear shear current are performed in the first time. The bifurcation pictures and two branches of solitary waves which are qualitatively similar to the irrotational capillary-gravity solitary waves are found. The bifurcation points are found to be monotonic with the shear strengths. Stability properties of these branches are tested using the numerical evolution equation based on the time-dependent conformal map technique. Finally, both head on collisions and overtaking collisions are computed mainly for the case of one elevation wave and one depression wave, which are firstly performed to the best of our knowledge.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the reviewers for providing them with constructive comments and suggestions. This work is supported by Open Fund (PLN1004) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).