The authors have declared that no competing interests exist.

As a result of mathematical modeling it has been shown that any closed electrical line can be interpreted as a ring waveguide where the Fermi-Pasta-Ulam recurrences of the electron and phonon currents interact with each other on the transversal and longitudinal periodical structures of the line conductor’s crystalline lattice as well as on the structures of the wire insulation. An electronic circuit simulating the mathematical model through the dynamics of magnons and phonons in a closed ferrite core with two different coils switched into the shoulders of a multivibrator has been developed. It has been demonstrated that the interacting ferromagnetic and ferroacoustic resonances excited simultaneously in a ferrite core qualitatively correspond to the dynamics of the electron and phonon currents interaction process in a closed electrical line.

In the earlier paper

First consider a closed electrical transmission line as a ring waveguide. We define the amplitude of electron current in the transversal plane of the waveguide as _{electron} and the amplitude of phonon current in the longitudinal plane of the waveguide as _{phonon}. Using the results of the paper

………. (1)

Where x -is the longitudinal coordinate along the waveguide and y –is the transversal coordinate across the waveguide, _{II, }_{⊥} - are the longitudinal and transversal spatial frequencies. _{1} and _{1 - }are the random functions reflecting the short and long wave heat oscillations along the conductor lattice correspondingly.

The interaction process between the two currents takes place both on a quantum scale

……(2)

The distance

…… (3)

Now equalizing the energies of electron and phonon waves and using a quantum ratio p = ℏ k, we can put:

……...(4)

Where

For a macro scale we can use the dispersion ratio for electromagnetic waves in a cylinder adjustable waveguide

And for longitudinal waves in a ring line the dispersion ratio looks like follows:

…….(6)

We can reduce the system (1) to the form in which the interaction process between electron and phonon currents taking place on both short and long spatial scales depends only on the temporal coordinate by means of introducing the time lags. So the new system looks like:

………...(7)

……..(8)

Where _{0}, _{2} are the periods of the time lags.

The functions z(t) and w(t) were introduced as time lag ones to unify the processes of dispersion on short and long scales reflecting the dissipation on the crystalline lattice and its non homogeneity:

For long scale (100-200steps) counting of the system (7) the expression for the frequency

whereas for shot scale (10 -20 steps) counting it looks like

since a small section of the transmission line in the transversal plane can be considered as a resonator excited by a number of sources consisting of vertical chains of atoms in the crystalline lattice

Computer study of the model system (7) shows that on a ring scale (100 steps and more) the dynamics of the currents demonstrates a resonant interaction between their Fermi-Pasta-Ulam recurrences

Computer study of the electric current model system (7) along a restricted part of the waveguide (10-20steps) shows a presence of the FPU recurrence only in the dynamics of the electron current (

Some more graphs were obtained during the study of the system (7) under different initial conditions for _{electron} and _{phonon}. For convenience of comparing with experimental oscillographs there are placed later in the paper.

The main purpose of experiment was to visualize the results of the theoretical model through a physical simulation of the system (7). First it was interesting to observe a resonant interaction between two FPU recurrences in the dynamics of the electron and phonon currents obtained as a result of the computer study of the system (7) on a long and short scales of counting (^{13} - 10^{9}Hz it was decided to use a closed ferromagnetic core with two different coils as a model of a transmission line. For that purpose there was used a commercial fly back transformer (

Two resonant processes of transformation of photons into magnons under a ferromagnetic resonance

……(9)

Where the density of the spin energy is

……..(10)

And the current density is:

….(11)

As it was shown

…… (12)

Where as a result of transformation of the vector to

the Serre-Frene vectors ^{13} Hz. That could be possible to provide a transformation of photons into magnons in a ferromagnetic. Any ferromagnetic has a magnetic moment which can have a certain precession. The frequency of such precession usually is a few megahertz. At the same time the frequency of the magnetic moment ω is the energy of a staying magnon

Eq. (13) means that under a ferromagnetic resonance a photon transforms into a magnon. The energies of a photon and a magnon are equal with the same impulse:

which gives a following equation:

Where ^{* }is the effective mass of a magnon:

The electron mass has been introduced into the (15) for better evaluation. So when T_{c} ≈ 10^{2} K

The magnon has an effective mass close to that of an electron. If to apply a magnetic field to the ferromagnetic and when μH = kT and H = 10^{4}oersted, then the magnon velocity V_{0} = (Ԑ_{0 }/ 2m^{*})^{1/2}will be V_{0} ≈ 3x10^{5} cm/sec

This evaluation means that it is possible to “observe” the picture of the non linear Shrodinger equation together with the FPU recurrence just by exciting in a ferrite core a transversal magnetic wave by an electric impulse in a coil having a small number of turns. It was realized by a symmetric multivibrator circuit (_{1}of 12 turns d=1mm wire on a ferrite core of the fly back transformer (

On the other hand the dynamics of the phonon current along the ring waveguide in the structure of the crystalline lattice by analogy with sound waves can be described within the framework of the Korteveg de Vries equation

Differently loaded both shoulders of the multivibrator (_{1} (

The research has resulted in developing a mathematical model of an electric current in a closed transmission line. The waveguide approach used in modeling pointed at a much more sophisticated nature of an electric current than it has been generally assumed. An electric current either AC or DC represents within the framework of the proposed model a complex resonant interaction process between the dynamics of the transversal electron current and longitudinal phonon current in the structure of a conductor as well as in an insulation layers. A physical model developed for evaluating possible solutions of the mathematical model allowed to propose a basis for constructing the FPU recurrence generators that were later used for medical purposes.

The problem of interaction of nonlinear resonances has always been a sophisticated one. The more complex proved to be the analysis of interaction between the spectra of the FPU recurrences

According to developed mathematical and physical models, a considerable regrouping of energy in favor of the electron current takes place in current transformers, in electrical motors, in the points of wire connections, in the structure of heating alloys whereas the phonon current is characteristic for lightning and for some Tesla’s long distance experiments when the frequency of generated electrical current lied in a band of hundreds of kilohertz. The electron current mode in a transmission line causes prevailing of a high frequency FPU recurrence not only in a conductor but in the insulation material as well, which could result in considerable damage of the electrical networks.

The author is thankful to the museum of Russian composer Piotr Chaykovsky in Klin Moscow Region which visiting has helped him to insight the idea of the electric current experimental model. The author is also thankful to the library of Princeton University for help.