25. Korteweg–de Vries type equations for waves propagating along the interface between air–water
Abstract: We present solutions of the water wave problem for a fluid layer of finite depth in the presence of gravity and surface tension. The method of multiple scale expansion is employed to obtain the Korteweg–de Vries (KdV) equations for solitons, which describes the behavior of the system for the free surface between air and water in a nonlinear approach. The solutions of the water wave problem split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as the solutions of the KdV equations. The solutions of the KdV equations are obtained analytically by using the tanh-function method. The dispersion relations of the model KdV equations are studied. Finally, we observe that the elevation of the water waves are in the form of traveling solitary waves. The horizontal and vertical velocities, and the phase diagrams of the velocity components have a nonlinear characters.
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