Continua of localized wave solutions via a complex similarity transformation

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Abstract
In the following, we obtain continua of localized wave solutions to the scalar homogeneous wave, damped wave, and Klein-Gordon equations. We do this by utilizing the fact that similar Ansätze (all of which involve a free-particle time-dependent Schrödinger-like equation) may be used to satisfy all three of these partial differential equations. This Schrödinger-like equation is reduced to an ordinary differential equation (ODE) using a dimensionless complex similarity transformation. A general solution to this ODE involving confluent hypergeometric functions is found. For an azimuthal dependence exp(iνφ),ν∈openR, this general solution includes many of the previously determined localized wave solutions as special cases.

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