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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.
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.
The independent coordinate has sense of time. Indeed, it may be proportional to time with some relation like , where is constant.
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.
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 period of pulsation, function is almost periodic.
where and are minimal and maximal values of ; this solution is written for the case when .
however, other solutions may be obtained using the translational invariance.
The ratio is a convenient parameter to characterize the amplitude of pulsation, then, the median value can be expressed as ; and the energy also is an elementary function of . For the case , an example of pulsation of the Toda oscillator is shown in Fig. 1.
In application, the quantity has no need to be physical energy of the system; in these cases, this dimensionless quantity may be called quasienergy.
The period of pulsation is increasing function of the amplitude .
At , the period
At , the period
In the whole range , the period and the frequency can be approximated with
with at least 8 significant figures. The relative error of this approximation does not exceed .
At small (but still positive) values of and , the pulsation decays slowly, and this decay can be described analytically. In the first approximation 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 the similar manner, as the period above. This allows to approximate the solution of the initial equation; and the error of such approximation is small compared to the difference between behavior of the idealized Toda oscillator and behavior of the experimental realization of the Toda oscillator as self-pulsing laser at the optical bench, although, qualitatively, a self-pulsing laser shows very similar behavior.3
The Toda chain equations of motion, in the continuous limit in which the distance between neighbors goes to zero, become the KdV equation.1 Here the index labeling the particle in the chain becomes the new spacial coordinate.
In contrast, the Toda field theory is achieved by introducing a new spacial 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.
- M. Toda (1975). "Studies of a non-linear lattice". Physics Reports 18 (1): 1. Bibcode:1975PhR....18....1T. doi:10.1016/0370-1573(75)90018-6.
- Politi, A. (1985). "Toda potential in laser equations". Zeitschrift fur Physik B 59 (1): 111–115. Bibcode:1985ZPhyB..59..111O. doi:10.1007/BF01325388. Text "lastOppo " ignored (help)
- 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. doi:10.1088/1751-8113/40/9/016.
- Kashaev, R.-M.; Reshetikhin, N. (1997). "Affine Toda field theory as a 3-dimensional integrable system". Commun.Math.Phys. 188: 251–266. doi:10.1007/s002200050164.