Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation

The equation of the surface waves in deep water, given here by (1), is extended to (2+1) dimensions, which is a novel equation.. It is shown that the surface waves equation is self- free source. So, it has a class of infinite solutions. Here many types of self-similar and semi-self similar solutions are obtained. The self-similar waves show various geometric structures. Among them, wave crest in the form of coupled lumps and soliton wave moving along the characteristic curve in the plane. It is entrained by troughs with cavities. The semi-self similar waves exhibit multi lumps or periodic waves with troughs and multi-periodic waves. The study of bifurcation shows that the trajectories are open, so that the traveling wave solutions are unstable. The time-dependent steepness-function is defined here and it is found that it attains a maximum value and then it decreases with time. The results found are interesting in ocean engineering and sciences. The extended unified method is used, here, to find the exact solutions, which was proposed recently.