# Invariant tori, action-angle variables, and phase space structure of the Rajeev-Ranken model

@article{Krishnaswami2019InvariantTA, title={Invariant tori, action-angle variables, and phase space structure of the Rajeev-Ranken model}, author={Govind S. Krishnaswami and T. R. Vishnu}, journal={Journal of Mathematical Physics}, year={2019} }

We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1+1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles the Neumann and Kirchhoff models, its equations may be viewed as the Euler equations for a centrally extended Euclidean algebra. The model has a Lax pair and r-matrix leading to four generically independent conserved quantities in… Expand

#### 2 Citations

Quantum Rajeev-Ranken model as an anharmonic oscillator

- Physics, Mathematics
- 2021

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,… Expand

An introduction to Lax pairs and the zero curvature representation

- Physics, Mathematics
- 2020

Lax pairs are a useful tool in finding conserved quantities of some dynamical systems. In this expository article, we give a motivated introduction to the idea of a Lax pair of matrices $(L,A)$,… Expand

#### References

SHOWING 1-10 OF 27 REFERENCES

On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

- Physics, Mathematics
- Journal of Physics Communications
- 2019

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the… Expand

Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method

- Physics
- 1978

A method is proposed for deriving and classifying relativistically invariant integrable systems that are sufficiently general to encompass all presently known two-dimensional solvable models, and for… Expand

Highly nonlinear wave solutions in a dual to the chiral model

- Physics
- 2016

We consider a two-dimensional scalar field theory with a nilpotent current algebra, which is dual to the Principal Chiral Model. The quantum theory is renormalizable and not asymptotically free: the… Expand

Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix

- Physics, Mathematics
- 1998

We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear… Expand

Symplectic structure of the moduli space of flat connection on a Riemann surface

- Mathematics, Physics
- 1995

We consider the canonical symplectic structure on the moduli space of flatg-connections on a Riemann surface of genusg withn marked points. Forg being a semisimple Lie algebra we obtain an explicit… Expand

Integrable Systems

- 2012

Many natural systems can be modelled by partial differential equations (PDEs), especially systems exhibiting wave-like phenomena. Such systems often have quantities that are conserved in time, common… Expand

Introduction to classical integrable systems

- Mathematics
- 2003

1. Introduction 2. Integrable dynamical systems 3. Synopsis of integrable systems 4. Algebraic methods 5. Analytical methods 6. The closed Toda chain 7. The Calogero-Moser model 8. Isomonodromic… Expand

Lectures on gauge theory and integrable systems

- Physics
- 1997

In these notes, I will describe the moduli space of flat connections on a principal bundle over a surface and its Poisson structure. I will then give examples of integrable systems on these spaces,… Expand

Separation of variables for the classical and quantum Neumann model

- Physics
- 1992

Abstract The method of separation of variables is shown to apply to both the classical and quantum Neumann model. In the classical case this nicely yields the linearization of the flow on the… Expand

Morse theory

- 1999

Topology on L a, b : Fix a broken λ λ , ... , λ . Its neighborhood consists of its deformations and smoothings. Key: every smooth trajectory has R symmetry. Can reduce to a level set in a Morse chart… Expand