Exact solutions of KdV equation with time-dependent coefficients

Some exact solutions of KdV equation with variable coefficients

“…We conclude that the solitary wave ansatz method proposed in [10,11] is a special case of the extended mapping transformation method.…”

Conservation laws and symmetries of time-dependent generalized KdV equations

“…(ii) The class of time-dependent generalized KdV equations (1) admits additional conservation laws given by low-order conserved densities (12) only for f (t, u) of the form (17a), (18a), (19a), (20a) (satisfying conditions (2)). The admitted conservation laws in each case are given by:…”

Possible management of near shore nonlinear surging waves through bottom boundary conditions

“…with the depth function D(X) and leakage velocity function g(X) are tuned as (49). Note that for decreasing depth D, which without leakage would make the wave amplitude to surge as in (46), due to the controlled tuning of the leakage the resultant solitonic wave function would suffer a damping of its amplitude as evident from (52). Moreover the solitonic wave flattens down with a change in its velocity along its propagation (see FIG 5).…”

“…which is a known integrable equation derivable from the hydrodynamic equations with cylindrical symmetry [35]. An exact solution of the variable coefficient KdV equation (62) is presented in [46] in the rational form as H = (c− 5 2 ξ) T . Using the relation with our original field: η 0 = D 2 9 H and reverting to our old coordinates ξ, X we can transform back the solution to obtain the required exact solution for the surface wave…”