On Reductions to the Second Painlevé Equation and N-Soliton Solutions of the Two and Three Dimensional Nonlinear Klein-Gordon Equations

The two and three-dimensional nonlinear Klein-Gordon (and Higgs) equations are reduced to the one-dimensional nonlinear Schrödinger equation. N-soliton solutions of the two and three-dimensional nonlinear Klein-Gordon (and Higgs) equations are obtained by substituting N-soliton solutions of the one-dimensional nonlinear Schrödinger equation into the similarity transformations. It is also shown that the two and three-dimensional Higgs equations can be reduced to the second Painlevé equation.