In physics, the Toda oscillator is a special kind of nonlinear oscillator. It represents a chain of particles with exponential potential interaction between neighbors.[1] These concepts are named after Morikazu Toda. The Toda oscillator is used as a simple model to understand the phenomenon of self-pulsation, which is a quasi-periodic pulsation of the output intensity of a solid-state laser in the transient regime.

Definition

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The Toda oscillator is a dynamical system of any origin, which can be described with dependent coordinate   and independent coordinate  , characterized in that the evolution along independent coordinate   can be approximated with equation

 

where  ,   and prime denotes the derivative.

Physical meaning

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The independent coordinate   has sense of time. Indeed, it may be proportional to time   with some relation like  , where   is constant.

The derivative   may have sense of velocity of particle with coordinate  ; then   can be interpreted as acceleration; and the mass of such a particle is equal to unity.

The dissipative function   may have sense of coefficient of the speed-proportional friction.

Usually, both parameters   and   are supposed to be positive; then this speed-proportional friction coefficient grows exponentially at large positive values of coordinate  .

The potential   is a fixed function, which also shows exponential growth at large positive values of coordinate  .

In the application in laser physics,   may have a sense of logarithm of number of photons in the laser cavity, related to its steady-state value. Then, the output power of such a laser is proportional to   and may show pulsation at oscillation of  .

Both analogies, with a unity mass particle and logarithm of number of photons, are useful in the analysis of behavior of the Toda oscillator.

Energy

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Rigorously, the oscillation is periodic only at  . Indeed, in the realization of the Toda oscillator as a self-pulsing laser, these parameters may have values of order of  ; during several pulses, the amplitude of pulsation does not change much. In this case, we can speak about the period of pulsation, since the function   is almost periodic.

In the case  , the energy of the oscillator   does not depend on  , and can be treated as a constant of motion. Then, during one period of pulsation, the relation between   and   can be expressed analytically: [2][3]

 

where   and   are minimal and maximal values of  ; this solution is written for the case when  .

however, other solutions may be obtained using the principle of translational invariance.

The ratio   is a convenient parameter to characterize the amplitude of pulsation. Using this, we can express the median value   as  ; and the energy   is also an elementary function of  .

In application, the quantity   need not be the physical energy of the system; in these cases, this dimensionless quantity may be called quasienergy.

Period of pulsation

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The period of pulsation is an increasing function of the amplitude  .

When  , the period  

When  , the period  

In the whole range  , the period   and frequency   can be approximated by

 
 

to at least 8 significant figures. The relative error of this approximation does not exceed  .

Decay of pulsation

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At small (but still positive) values of   and  , the pulsation decays slowly, and this decay can be described analytically. In the first approximation, the parameters   and   give additive contributions to the decay; the decay rate, as well as the amplitude and phase of the nonlinear oscillation, can be approximated with elementary functions in a manner similar to the period above. In describing the behavior of the idealized Toda oscillator, the error of such approximations is smaller than the differences between the ideal and its experimental realization as a self-pulsing laser at the optical bench. However, a self-pulsing laser shows qualitatively very similar behavior.[3]

Continuous limit

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The Toda chain equations of motion, in the continuous limit in which the distance between neighbors goes to zero, become the Korteweg–de Vries equation (KdV) equation.[1] Here the index labeling the particle in the chain becomes the new spatial coordinate.

In contrast, the Toda field theory is achieved by introducing a new spatial coordinate which is independent of the chain index label. This is done in a relativistically invariant way, so that time and space are treated on equal grounds.[4] This means that the Toda field theory is not a continuous limit of the Toda chain.

References

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  1. ^ a b Toda, M. (1975). "Studies of a non-linear lattice". Physics Reports. 18 (1): 1. Bibcode:1975PhR....18....1T. doi:10.1016/0370-1573(75)90018-6.
  2. ^ Oppo, G.L.; Politi, A. (1985). "Toda potential in laser equations". Zeitschrift für Physik B. 59 (1): 111–115. Bibcode:1985ZPhyB..59..111O. doi:10.1007/BF01325388. S2CID 119657810.
  3. ^ a b Kouznetsov, D.; Bisson, J.-F.; Li, J.; Ueda, K. (2007). "Self-pulsing laser as Toda oscillator: Approximation through elementary functions". Journal of Physics A. 40 (9): 1–18. Bibcode:2007JPhA...40.2107K. CiteSeerX 10.1.1.535.5379. doi:10.1088/1751-8113/40/9/016. S2CID 53330023.
  4. ^ Kashaev, R.-M.; Reshetikhin, N. (1997). "Affine Toda field theory as a 3-dimensional integrable system". Communications in Mathematical Physics. 188 (2): 251–266. arXiv:hep-th/9507065. Bibcode:1997CMaPh.188..251K. doi:10.1007/s002200050164. S2CID 17196702.