ON THE FLUCTUATIONS OF WATER WAVES GOVERNED BY THE CAMASSA-HOLM AND KdV EQUATIONS IN (1+1)-DIMENSION

In this paper, we are planning to consider the fluctuations of two nonlinear equations which govern the dynamics of water waves named Camassa-Holm and KdV. We consider the total number of positive slopes N-tot(+) produced when the fluctuations of the wave velocity u(x) of a surface wave of a fluid, for example water, is crossed by the level (u(x) - ((u) over bar)) = alpha in the Camassa-Holm and KdV equations. Here, we just concentrate on the high Reynolds number limit and do the level crossing analysis where v -> 0. In our desired limit, the dissipative term becomes absent or very weak compared to the nonlinear term which is responsible for increasing the amplitude and creating wave steepening, which results in the appearance of shocks. Thus, our analysis works at the times before the appearance of shocks. Our aim in this paper is to show how the quantity, v(alpha)(+), counts the fluctuations of the wave velocity in the surface water wave fluctuations which are governed by the KdV and Camassa-Holm (CH) equations.