Dynamical symmetries of semi-linear Schrödinger and diffusion equations

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Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra conf₃. We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of conf₃ are classified and the complete list of conditionally invariant semi-linear Schrödinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed.