On asymptotically equivalent shallow water wave equations

The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution

Approximate analytical solutions of KdV and burgers' equations via HAM and nHAM

This article presents a comparative study of the accuracy between homotopy analysis method (HAM) and a new technique of homotopy analysis method (nHAM) for the Korteweg-de Vries (KdV) and Burgers' equations. The resulted HAM and nHAM solutions at 8th-order and 6th-order approximations are then compared with that of the exact soliton solutions of KdV and Burgers' equations, respectively. These results are shown to be in excellent agreement with the exact soliton solution. However, the result of HAM solution is ratified to be more accurate than the nHAM solution, which conforms to the existing findin

Approximate Analytic Solution for the KdV and Burger Equations with the Homotopy Analysis Method

The homotopy analysis method (HAM) is applied to obtain the approximate analytic solution of the Korteweg-de Vries (KdV) and Burgers equations. The homotopy analysis method (HAM) is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. HAM contains the auxiliary parameter ħ, which provides us with a straightforward way to adjust and control the convergence region of the series solution. The resulted HAM solution at 8th-order and 14th-order approximation is then compared with that of the exact soliton solutions of KdV and Burgers equations, respectively, and shown to be in excellent agreement