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Solitons Description of the Model

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We would like to introduce to you a mechanical model for the sine-Gordon equation. This equation admits "soliton solutions" among others and we will be mainly concerned with these. Our model consists of a ribbon of infinite length which has been marked on both sides. The upper border is firmly connected to a wire. From it, the ribbon is hanging down like a sock from a clothesline. The lower edge is made of a massive sheet which is acted on by gravity. Otherwise, the structure is massless. The ribbon can also be twisted. The wire in the middle is then also twisted and reacts to it with a repulsive force depending linearly on the twisting angle phi. This angle itself depends on the location on the wire, x, and on the time t; thus phi is a function of x and t. It satisfies the sine-Gordon equation. Because of the nonvanishing gravity g this equation also contains the sine of the angle phi and is therefore nonlinear.

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Solution for Small Twisting Angles

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For small twisting angles phi the nonlinear sine-Gordon equation becomes a linear equation. Namely in this case, the sine of phi may be replaced by the angle phi itself. This excitation, which is localized at

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the beginning, spreads since, due to the nonvanishing gravity g, the linearized equation shows dispersion. We will not demonstrate the well-known dispersion-free soliton-solitions for g equal to zero.

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Constant in Time Soliton

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This is a solution of the sine-Gordon equation which is constant in time. We call the left-handed screw a soliton. On a finite stretch, the ribbon is turned around once almost completely and is everywhere else hanging almost straight down.

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Soliton Moves to Right

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The soliton is also capable of travelling along the ribbon.

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This solution proceeds with constant velocity and without changing its shape from front to back.

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Soliton Moves to left

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This soliton moves into the opposite direction. Solitons are generally defined as excitations which travel without changing their shape and survive encounters with other solitons unaltered. It is remarkable that this kind of solution exists at all for nonlinear differential equations.

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Constant in Time Antisoliton

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This solution, being once again constant in time, is a right-handed screw and thus not a soliton as defined above. We call it an "antisoliton". Since reversing a screw does not change its sense of rotation,

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looking from behind at a soliton one would still see a left-handed screw and not an antisoliton.

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Antisoliton Moves to Right

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The antisoliton or right-handed screw, just as the soliton, can travel on the ribbon without changing its shape.

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Antisoliton Moves to Left

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Since the sense of rotation for soliton and antisoliton is opposite, they also turn the ribbon into opposite directions when they are travelling.

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Lorentz Contraction to Solitons

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The sine-Gordon equation is a relativistic equation and therefore shows relativistic effects such a maximal velocity - analogous to the velocity of light - beta = 1 and Lorentz contraction.

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We compare the extension of a soliton at rest, beta = 0, to that of a soliton moving at beta = 0.3. For a fast soliton with beta = 0.9, the contraction is evident. Solitons cannot reach the limiting velocity beta = 1.

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Encounter of Two Solitons

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In this scene, two solitons having equal and opposite velocities meet. Before they encounter, each of them behaves as if the other does not exist. As they approach each other, they try to turn the ribbon between them into opposite directions. Thus the solitons repel; they stop, start to run backward and leave again. Once again the same event, looked at more closelv. The solitons do not change very much during their encounter and after it there still are two solitons, moving in opposite directions. The same once again. The additional lines at the

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beginning have the same speed as the two solitons and do not change it during the encounter. Afterwards it appears as if hardly anything had happened: Which each of the lines a soliton moves along, perhaps displaced a little. The same occurrence as seen from the point of view of someone riding along with one

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soliton. Initially this soliton is at rest. During the encounter it is hit by the other soliton and starts to move while the other soliton comes to rest. The soliton which was previously at rest moves on. Thus the solitons behave just as two masses which are scattered elastically do.

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Encounter of Soliton and Antisoliton

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This is an encounter of a soliton - coming from the right - and an antisoliton. They turn the ribbon between them in the same direction and thus attract and penetrate each other. After the complicated interaction, there is still a soliton, proceeding to the

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left, and an antisoliton, proceeding to the right. We also look at his occurrence more closely. It is a very remarkable fact that soliton and antisoliton both survive this violent encounter undamaged. We have once again added the two vertical lines. They

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show that, at the end, the interaction has influenced the two excitations very little: they still move with the same speed and in the same direction as before, perhaps displaced a little bit.

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Finally the same occurrence as seen from the soliton's rest frame. In a complicated interaction, the antisoliton passes the soliton and, after that, the soliton is still at rest and the antisoliton is still moving to the right with the same velocity as before.

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A soliton at rest is approached from the left by two solitons, one after each other; at the end the last one has stopped and the other two are moving ahead to the right.

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Two Solitons and One Antisoliton

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This complicated-looking encounter of two solitons and one antisoliton can however be decomposed into a sequence of already well-known interactions of only two excitations. It is very remarkable indeed that not only do solitary solutions of the nonlinear sine-Gordon equation not change their shape in time but, furthermore, survive violent and complicated collisions unhurt and without change of velocity!

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This solution of the sine-Gordon equation is called a breather for obvious reasons. It corresponds to a bound state of a soliton and an antisoliton which do not have enough energy to separate. Vertical lines indicate the extension of this breather which is at rest.

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For a breather in motion we expect Lorentz-contraction. Since the shape of a breather constantly changes, contraction is obvious only for a fast-moving one.

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Collision of Soliton and Breather

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Breathers, solitons and antisolitons can collide in countless combinations. This is,for example, a collision between a breather and a soliton. After the encounter, both continue to proceed with unchanged velocity. In particular, the breather has not been broken up into its parts, soliton and antisoliton.

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Collision of Two Breathers

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This is our second example of a complicated interaction: two breathers collide. After the collision, neither has been broken up.

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Decay of Pulse into Soliton and Antisoliton

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The hammer blow indicats a special excitation which decays into soliton and antisoliton. For impulse-like excitations, decays of this type are typical.

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Infinite Numbers of Solitons with Like Velocities.

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If any finite number of solitions larger than one is excited at the same time, not all of them can have the same speed; at some time, there has been or will be a collision. The total structure then changes its shape and thus is not a soliton-solution. An infinite number of solitons however can proceed along the ribbon with uniform velocity and without any change of shape. Since - just a plane wave on a linear system - this solution is extended infinitely, it is not a soliton solution. During recent years, solitons have been applied in many branches of physics. Some important examples are: Water waves, Josephson-junctions, plasmaphysics, interaction of radiation with plasma, propagation of Bloch walls in ferromagnets, propagation of crystal dislocations, self-induced transparency, transmission of nerve lines, models for elementary particles.