Dear Colleagues! My Name Is NN. I Will Present the Team of Russia with the Problem Soliton

Dear colleagues! My name is NN. I will present the team of Russia with the problem Soliton.

The problem says: A chain of similar pendula is mounted equidistantly along a horizontal axis, with adjacent pendula being connected with light strings. Each pendulum can rotate about the axis but can not move sideways. We are proposed to investigate the propagation of a deflection along such a chain and to answer the question: What is the speed for a solitary wave, when each pendulum undergoes an entire 360º revolution?

Wave propagation in a chain of coupled pendulawas comprehensivelystudied by many authors. On this pictureyou can see the demo setup of Scott. It consist of pendula on horizontal axis, coupled by axial springs.

This is our experimental setup.In accordance with the condition of the problem pendula are coupled with a pre-tensioned rubber band at their ends.

This is the list of main parameters of our setup.

On this video you can see asolitary wave, propagating along the chain of pendula.

As already mentioned, the literature is usually considered a model in which the pendulums are coupled by axial torsion springs.

Each pendulum experiences the angular moment due to the gravity and the angular moments from two torsion springs, which are proportional to the difference in angles of rotation of the coupled pendulums.

Writing the Newton’s second law for a pendulum, we obtain this nonlinear difference equation. For wave processes which change only slightly over a distance Δ, this equation can be written as a partial differential equation,which called sine-Gordon.

Note two limiting cases of sine-Gordon equation. If there is no interaction between pendula, this equation converts into the pendulum equation. If gravity tends tozero, it converts into thewave equation.

However, in our problem the pendula must be coupled with the end strings.

The main difference is thatin this case the torque is not proportional to the angle.

Using elementary geometry, we got this complicated expression for the torque. Let’s plot a graph for this function for the parameters of our experimental setup.

To our luck, this graph is very close to linear. Hence we can approximate itas a linear and use sine-Gordon equation. The wave velocity с in this equation in our case is about 2.5 m/s.

First of all, let’s consider small-amplitude waves.

If the variation of α is restricted to small angular variations about zero, we have linear Klein-Gordon equation instead of nonlinear sine-Gordon equation. The wave solution is found in this form. Herekisa circular wavenumber and ωis a circular frequency.

The dispersion relation shows howωdepends onk.When wavenumbersarelarge and waves are short, ω is proportional tok. When wavenumbersare smalland waves are long, ω is close toΩ, which is the frequency of pendulum small oscillations. Waves with frequency less then Ω in this medium can not propagate.

To measure dispersionwe used the wave generator.

This diagram shows our experimental results. We observe a good agreement with the theoretical prediction.Notice that the velocity of short waves is about 2.5 m/s, which coincides with the result of previous calculation.

Now let me consider the soliton solution of sin-Gordon equation.

Soliton is a solitary wave which propagates in a nonlinear medium.Soliton can travel over very large distances without changing its shape.

A simple class of sine-Gordon solutions are travelling waves of permanent profile which propagate at a constant speed.

Soliton is a special case of a travelling wave. In this case on both sides far from the disturbance the pendula hang down and do not swing.

In a single sine-Gordon solution the angle depends on the time and the longitudinal coordinate as follows. Such a solution can travel at any velocity less then c.

This is a graph for the soliton solution. If the pendula are rotating clockwise it is called kink. Otherwise it is antikink.

Soliton case has the phenomenon of dispersion as well. It is described by this formula. Stationary solitons are the longest. The higher the velocity, the shorter the core of the disturbance.

We made an experiment to prove this theory.

We measured the length of the soliton as a distance between two horizontal pendula. This stationary soliton has the length about 13 pendula.

This graph presents how the length of soliton depends on its velocity. As you can see,experimental results have a great agreement with theoretical prediction. Of course, we are not able to run very short solitons, since rubber bands already stretched very tight.

Also we made a computer simulation.

The pendula are connected with each other by invisible end strings. In this model there is not any pre-tension of the end strings.

This graph presents how the length of soliton depends on its velocity. You can see a good agreement between the theory and the computer simulationagain.

Let me summarize our work.

The system of pendula with end strings generally behaves the same way as the system of pendula with axial torsional springs. So we can use sine-Gordon equation for its description.

The length of the sine-Gordon soliton depends on its velocity. The stationary soliton has the largest length. The faster moving soliton, the shorter its length. Limiting velocity of the soliton is equal to the speed of short waves of small amplitude.

This is a list of references we used.

Thank you for your attention!