Symmetry Reduction and Exact Solutions of the (3+1)-DimensionalBreaking Soliton Equation

By means of the generalized direct method, a relationship isconstructed between the new solutions and the old ones of the(3+1)-dimensional breaking soliton equation. Based on therelationship, a new solution is obtained by using a givensolution of the equation. The symmetry is also obtained for the(3+1)-dimensional breaking soliton equation. By using the equivalentvector of the symmetry, we construct a seven-dimensional symmetryalgebra and get the optimal system of group-invariant solutions. Toevery case of the optimal system, the (3+1)-dimensional breakingsoliton equation is reduced and some solutions to the reducedequations are obtained. Furthermore, some new explicit solutions arefound for the (3+1)-dimensional breaking soliton equation.