Particle trajectories under interactions between solitary waves and a linear shear current

This paper is concerned with particle trajectories beneath solitary waves when a linear shear current exists. The fluid is assumed to be incompressible and inviscid, lying on a flat bed. Classical asymptotic expansion is used to obtain a Korteweg-de Vries (KdV) equation, then a forth-order Runge-Kutta method is applied to get the approximate particle trajectories. On the other hand, our particular attention is paid to the direct numerical simulation (DNS) to the original Euler equations. A conformal map is used to solve the nonlinear boundary value problem. Highaccuracy numerical solutions are then obtained through the fast Fourier transform (FFT) and compared with the asymptotic solutions, which shows a good agreement when wave amplitude is small. Further, it also yields that there are different types of particle trajectories. Most surprisingly, periodic motion of particles could exist under solitary waves, which is due to the wave-current interaction.