Pl. Overfelt, CONTINUA OF LOCALIZED WAVE SOLUTIONS VIA A COMPLEX SIMILARITY TRANSFORMATION, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 47(6), 1993, pp. 4430-4438
In the following, we obtain continua of localized wave solutions to th
e scalar homogeneous wave, damped wave, and Klein-Gordon equations. We
do this by utilizing the fact that similar Ansatze (all of which invo
lve a free-particle time-dependent Schrodinger-like equation) may be u
sed to satisfy all three of these partial differential equations. This
Schrodinger-like equation is reduced to an ordinary differential equa
tion (ODE) using a dimensionless complex similarity transformation. A
general solution to this ODE involving confluent hypergeometric functi
ons is found. For an azimuthal dependence exp(inuphi), nu is-an-elemen
t-of R, this general solution includes many of the previously determin
ed localized wave solutions as special cases.