Single soliton solution to the extended KdV equation over uneven depth

Abstract

Abstract. In this note we look at the influence of a shallow, uneven riverbed on a soliton. The idea consists in an approximate transformation of the equation governing wave motion over an uneven bottom to an equation for a flat one for which the exact solution exists. The calculation is one space dimensional, and so corresponds to long trenches or banks under wide rivers or oceans.

The New Extended KdV Equation for the Case of an Uneven Bottom

“…The creation and then detachment of the small amplitude wave packet in front of the main wave is clearly exposed in the insert. This is qualitatively the same feature as observed in our previous papers [1,2,10] for wave motion according to the erroneous equation [2, Eq. ( 18)].…”

“…The creation and then detachment of the small amplitude wave packet in front of the main wave is clearly exposed in the insert. This is qualitatively the same feature as observed in our previous papers [1,2,10] for wave Quantitatively the effect has much smaller amplitude, for realistic values of parameters α, β, δ it is smaller than 1% of the solitons amplitude. On the other hand, even such small effect suggests the origin of the very tiny wrinkles observed always on the water surface at the seashore.…”

“…Since the equation ( 31) is valid only for the piecewise linear bottom function h(x) = kx the trapezoidal bottom is allowed. The size and location of the trapezoid allows us also to compare the results with those presented in [10]. Note that the bottom function is drawn not in scale.…”

New Exact Superposition Solutions to KdV2 Equation

“…Later (2) was derived in a different way in [13] and as a by-product in derivation of the equation for waves over uneven bottom in [14,15]. Therefore exact solutions of (2) are the best initial conditions for performing numerical evolution of waves entering the regions where the bottom changes, see [16].…”