NONCLASSICAL SYMMETRY REDUCTIONS FOR THE DISPERSIVE WAVE-EQUATIONS IN SHALLOW-WATER

All (six types of) the similarity reductions of the dispersive wave equations in shallow water are obtained by the direct method. The group explanation of the results is also given. It is pointed out that, in order to find all the results by nonclassical Lie approach, two additional condition equations should be satisfied at the same time together with two original equations. The logarithmic branch points for time t can be included in two types of the similarity solutions, though the model can be cast into the canonical Hamiltonian form.