Properties of the Magma and Modified Magma Equations

Solutions of the “magma equation”, f_t=[fⁿ{(f^-mf_t)_x-1}]_x, are investigated for compact support initial conditions. It is shown analytically that traveling wave solutions with compact support do not exist for this equation. Numerical integration demonstrates that although initial conditions with compact support produce solitary waves, these solitary waves do not interact like solitons. Moreover, a relationship between the magma equation and the KdV equation is established, and this relationship is used to explain previously observed soliton-like behavior. Both equations are shown to be related to the modified magma equation, F_T+[F³(3F_XX+1)]_x=0, which is proven to have N-soliton solutions with compact support and an infinite number of conserved densities.