Invariances, conservation laws and conserved quantities of the two-dimensional nonlinear Schrodinger-type equation

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014.Symmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts