Group theoretical methods for solving multidimensional nonlinear partial differential equations

Sammanfattning: New methods for constructing both exact and approximate solutions of multidimensional nonlinear partial differential equations are developed. The basis of the methods is taken from the classical Lie transformation group theory. New symmetry concepts are investigated for the construction of solutions of important nonlinear partial differential equations. Those concepts are: conditional symmetries, hidden symmetries, and approximate symmetries. Moreover, we propose a method by which a multidimensional wave equation can be related and classified with respect to a compatible system of multidimensional differential equations. The equations under investigation are: nonlinear Schrödinger equations, nonlinear heat equations, a generalized van der Pol equation, and nonlinear d'Alembert equations. The new solutions presented in this thesis could be of fundamental importance in the applications of the physical processes modelled by the above mentioned differential equations.

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