The Rajeev-Ranken model is a mechanical system with 3 degrees of freedom describing screw-type nonlinear waves (screwons) in a scalar field theory dual to the 1+1D SU(2) Principal Chiral Model. This scalar field is strongly coupled in the UV and could serve as a toy-model to study nonperturbative features of theories with a perturbative Landau pole.

The system admits Hamiltonian formulations based on compatible degenerate nilpotent and Euclidean Poisson brackets, Lax pairs and r-matrices and is shown to be Liouville integrable. The phase space is foliated by invariant tori that arise as common level sets of conserved quantities. A family of action-angle variables is obtained. 

Darboux coordinates are used to interpret the model as a novel 3D cylindrically symmetric quartic oscillator with a rotational energy. The Schrodinger equation is solvable to the extent that variables may be separated. Strong and weak coupling limits, as well as numerical diagonalization, are employed to understand the screwon spectrum. We find a 2/3-rds power-law dispersion relation for quantized screwons and a Poissonian level spacing distribution. Finally, our quantization is used to find an infinite-dimensional unitary representation of the underlying step-3 nilpotent Lie algebra.