The non-linear differential equation and the oscillation frequency were solved using the Jacobi elliptie function and complete elliptic integral of the first kind and the particle spatial and momentum distribut ions were derived from phase space density.
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J. A. Ellison , Phys. Rev., B 12 , 11 , 4771 ( 1975 ).
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P. F. Byrd and M. D. Friendman , Handbook of elliptic integrals for engineers and physicists , Springer-Verlag , 1954 .