Integrability and dynamics of the Rajeev-Ranken model

Vishnu, T. R.

doi:10.48550/arxiv.2109.12579

2021

DOI: 10.48550/arxiv.2109.12579

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Integrability and dynamics of the Rajeev-Ranken model

T. R. Vishnu¹

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“…§7 contains a discussion of our results and some open questions. Some of the results in this paper have appeared in the PhD thesis [15].…”

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confidence: 92%

Quantum Rajeev-Ranken model as an anharmonic oscillator

Krishnaswami,

Vishnu

2021

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The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schrödinger equation, we find that the radial equation has a four-term recurrence relation. It is of type [0, 1, 1 6 ] and lies beyond the ellipsoidal Lamé and Heun equations in Ince's classification. At strong coupling λ , the energies of highly excited states are shown to depend on the scaling variable λk . The energy spectrum at weak coupling and its dependence on wavenumber k in a double-scaling strong coupling limit are obtained. The semi-classical WKB quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a (λk) 2/3 dispersion relation for highly energetic quantized 'screwons' at moderate and strong coupling. We also suggest a mapping between our radial equation and one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.

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“…§7 contains a discussion of our results and some open questions. Some of the results in this paper have appeared in the PhD thesis [15].…”

mentioning

confidence: 92%

Quantum Rajeev-Ranken model as an anharmonic oscillator

Krishnaswami,

Vishnu

2021

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Integrability and dynamics of the Rajeev-Ranken model

Cited by 1 publication

References 14 publications

Quantum Rajeev-Ranken model as an anharmonic oscillator

Quantum Rajeev-Ranken model as an anharmonic oscillator

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