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Elliptic Solutions, Recursion Operators and Complete Lie-Backlund Symmetry for the Harry-Dym Equation

We have made an exhaustive analysis for the equation σt = σ3σκκκ - ασ3σκ, which in the special case reduces to the Harry-Dym equation. First of all we have deduced the Lie point symmetries and the corresponding ordinary differential equation, through the similarity forms. Next the extended Lie-Backlund type generators are deduced. Our equation possesses the special feature of having explicit σ dependent function in the coefficient of highest derivative term, in contrast to the cases discussed by Fokas et al. From the structure of ηI, ηII, ηIII it is easy to deduce the recursion operator Δ. In the second part we deal with the cnoidal wave like solutions. From the Fourier spectrum analysis we show that a cnoidal wave breaks into several ordinary solitary waves.