Toda Oscillator
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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the Toda oscillator is a special kind of nonlinear oscillator. It represents a chain of particles with exponential potential interaction between neighbors. These concepts are named after
Morikazu Toda was a Japanese physicist, best known for the discovery of the Toda lattice. His main interests were in statistical mechanics and condensed matter physics. Career After graduating from the Department of Physics, Tokyo University he became associ ...
. The Toda oscillator is used as a simple model to understand the phenomenon of
self-pulsation Self-pulsation is a transient phenomenon in continuous-wave lasers. Self-pulsation takes place at the beginning of laser action. As the pump is switched on, the gain in the active medium rises and exceeds the steady-state value. The number of ph ...
, which is a quasi-periodic pulsation of the output intensity of a
solid-state laser A solid-state laser is a laser that uses a active laser medium, gain medium that is a solid, rather than a liquid as in dye lasers or a gas as in gas lasers. Semiconductor-based lasers are also in the solid state, but are generally considered as ...
in the transient regime.


Definition

The Toda oscillator is a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
of any origin, which can be described with dependent coordinate ~x~ and independent coordinate ~z~, characterized in that the
evolution Evolution is the change in the heritable Phenotypic trait, characteristics of biological populations over successive generations. It occurs when evolutionary processes such as natural selection and genetic drift act on genetic variation, re ...
along independent coordinate ~z~ can be approximated with equation : \frac+ D(x)\frac+ \Phi'(x) =0, where ~D(x)=u e^+v~, ~\Phi(x)=e^x-x-1~ and prime denotes the derivative.


Physical meaning

The independent coordinate ~z~ has sense of
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
. Indeed, it may be proportional to time ~t~ with some relation like ~z=t/t_0~, where ~t_0~ is constant. The
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
~\dot x=\frac may have sense of
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
of particle with coordinate ~x~; then ~\ddot x=\frac~ can be interpreted as
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
; and the mass of such a particle is equal to unity. The dissipative function ~D~ may have sense of coefficient of the speed-proportional
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
. Usually, both parameters ~u~ and ~v~ are supposed to be positive; then this speed-proportional friction coefficient grows exponentially at large positive values of coordinate ~x~. The potential ~\Phi(x)=e^x-x-1~ is a fixed function, which also shows exponential growth at large positive values of coordinate ~x~. In the application in
laser physics Laser science or laser physics is a branch of optics that describes the theory and practice of lasers. Laser science is principally concerned with quantum electronics, laser construction, optical cavity design, the physics of producing a po ...
, ~x~ may have a sense of
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
of number of photons in the
laser cavity An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that confines light waves similarly to how a cavity resonator confines microwaves. Optical cavities are a major component of lasers, ...
, related to its steady-state value. Then, the output power of such a laser is proportional to ~\exp(x)~ and may show pulsation at
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of ~x~. 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

Rigorously, the oscillation is periodic only at ~u=v=0~. Indeed, in the realization of the Toda oscillator as a self-pulsing laser, these parameters may have values of order of ~10^~; during several pulses, the amplitude of pulsation does not change much. In this case, we can speak about the
period Period may refer to: Common uses * Period (punctuation) * Era, a length or span of time *Menstruation, commonly referred to as a "period" Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (o ...
of pulsation, since the function ~x=x(t)~ is almost periodic. In the case ~u=v=0~, the energy of the oscillator ~E=\frac 12 \left(\frac\right)^+\Phi(x)~ does not depend on ~z~, and can be treated as a constant of motion. Then, during one period of pulsation, the relation between ~x~ and ~z~ can be expressed analytically: : z=\pm\int_^\!\!\frac where ~x_~ and ~x_~ are minimal and maximal values of ~x~; this solution is written for the case when \dot x(0)=0. however, other solutions may be obtained using the principle of
translational invariance In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation. Analogously, an oper ...
. The ratio ~x_\max/x_\min=2\gamma~ is a convenient parameter to characterize the amplitude of pulsation. Using this, we can express the median value \delta=\frac as \delta= \ln\frac ; and the energy E=E(\gamma)=\frac+\ln\frac-1 is also an
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...
of ~\gamma~. In application, the quantity E need not be the physical energy of the system; in these cases, this dimensionless quantity may be called quasienergy.


Period of pulsation

The period of pulsation is an increasing function of the amplitude ~\gamma~. When ~\gamma \ll 1~, the period ~T(\gamma)=2\pi \left( 1 + \frac + O(\gamma^4) \right) ~ When ~\gamma \gg 1~, the period ~T(\gamma)= 4\gamma^ \left(1+O(1/\gamma)\right) ~ In the whole range ~\gamma > 0~, the period ~~ and frequency ~k(\gamma)=\frac~ can be approximated by : k_\text(\gamma)= \frac = : \left( \frac \right)^ to at least 8
significant figures Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcom ...
. The
relative error The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation derived for it. This inherent error in approximation can be quantified and express ...
of this approximation does not exceed 22 \times 10^ .


Decay of pulsation

At small (but still positive) values of ~u~ and ~v~, the pulsation decays slowly, and this decay can be described analytically. In the first approximation, the parameters ~u~ and ~v~ 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.


Continuous limit

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. Here the index labeling the particle in the chain becomes the new spatial coordinate. In contrast, the
Toda field theory In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Lie algebra and a specific Lagrangian. Formulation Fixing the Li ...
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. This means that the Toda field theory is not a continuous limit of the Toda chain.


References

{{reflist Mathematical physics Atomic, molecular, and optical physics