Toda Oscillator - Energy

Energy

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.

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

z=\pm\int_x^{x_\max}\!\!\frac{{\rm d}a} {\sqrt{2}\sqrt{E-\Phi(a)}}

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 \delta=\frac{x_\max -x_\min}{1} can be expressed as \delta= \ln\frac{\sin(\gamma)}{\gamma} ; and the energy E=E(\gamma)=\frac{\gamma}{\tanh(\gamma)}+\ln\frac{\sinh \gamma}{\gamma}-1 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.

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