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About a Discrete Cellular Soliton (computer simulation)

For the first time a mathematical object is presented - a reversible cellular Automaton - with many paradoxical qualities, the main ones among them are: a frequent quickly return to its original state, the presence of a large number of conservation laws and paradoxical"fuzzy"symmetries, which connects the current position of the automaton with its signature Main Integral.


Introduction
In 1980, the journal Science published an article of Feigenbaum M. J. Universal Behavior in Nonlinear Systems [1]. This article was a simple mathematical observation concerning one-dimensional maps. Of particular interest in this situation was the fact that the observation was made by simply using a calculator! And over time this work has become a progenitor of serious mathematical sciences. Now, in comparison with 1980, the situation has changed dramatically. Home computers have appeared. This allows us to do much more serious and profound observations. We are presenting one of them. We called it the phenomenon of Discrete Cellular Soliton (DCS). It refers to the reversible cellular automata, and so first let's talk what it is.

What is cellular automaton and reversible cellular automaton?
Let`s take the definition of cellular automaton from the Wikipedia.
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off" (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its neighborhood (usually including the cell itself) is defined relative to the specified cell. For example, the neighborhood of a cell might be defined as the set of cells a distance of 2 or less from the cell. An initial state (time t=0) is selected by assigning a state for each cell. A new generation is created (advancing t by 1), according to some fixed rule (generally, a mathematical function) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood. For example, the rule might be that the cell is "On" in the next generation if exactly two of the cells in the neighborhood are "On" in the current generation, otherwise the cell is "Off" in the next generation. Typically, the rule for updating the state of cells is the same for each cell and does not change over time, and is applied to the whole grid simultaneously, though exceptions are known.
Everything is clear. But we introduce our little pre-determination. Let`s call "the set of cells, called neighborhood" -the mask of a cellular automaton. Let`s cover the mask with a minimum square. Let`s call his side -k. Then let`s call the quantity [(k-1)/2] -a rank of the mask. And then we give an example, taken from the same source -from the Wikipedia. The Game of Life.
The universe of the Game of Life is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, alive or dead. Every cell interacts with its eight neighbours, which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur: 1. Any live cell with fewer than two live neighbours dies, as if caused by under-population. 2. Any live cell with two or three live neighbours lives on to the next generation. 3. Any live cell with more than three live neighbours dies, as if by overcrowding. 4, Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction. Then in the Wikipedia it is noted that these simple rules lead «to a huge variety of shapes that can occur in the game». It is practically a common quality of many CA! Despite the fact that the laws of the transition and the initial conditions can be very simple, the behavior of the CA can be extremely difficult and unusual. So, it all takes place in today's science as "Cognition of Difficult things" and there is no, even the similarity of the general theory.
But we will talk about a certain subclass of the CA about which «at least something is known». In the book of Margopolus Toffoli [2] they are called as reversible cellular automata (RCA) of the second order. Since there is a "second" order, then there must be a "first". But we'll talk exactly about such RCA, and in our own signs. This RCA has three state of the cell, and its structure is exhaustively illustrated in figure 2. We shall remember well that the cells A that we have are «white», the cells B are «blue», and the cells C are the «red»! What is "known in advance" about it and why is it interesting? It is interesting that if you make the so-called transliteration -that is you replace all the cells B with C and all cells C with B, and, further, apply the same rules of transition... then the automaton will go back in time! (See [2], though it is obviously).
And if it is so, it means that any state of the cellular automaton always has exactly one previous, and it can be found easily! (To make a transliteration, and then to make a move back).
And if it is so, it means that such a cellular automaton will always go on its own Cycle, not overlapping with others, and if the total number of states is finite, by all means it will pass through its starting point some time.
After what period of time will the cellular automaton do it? Usually, if the Automaton does not have an obvious symmetry, it will do it after a very, very long period of time, the order of the total number of states: 3 in the degree of the total number of cells. And if the size of the area is 100 x 100, so it is number 3 in the degree of 100x100 = 10000. It is transcendentally great! Therefore, we will call a Cycle with such a number of steps a Giant Cycle.  Figure 3 shows the typical movement of a random RCA. The way how Automaton goes to the Global Cycle. (His law is not given, and it is not necessary!) We started with two cells B, the rest -A, and after only 180 steps we got seething mess. One can say that we have visualized, what it means -number 3 in the degree of 10000.
And from the reason of simple common sense it follows that in general ANY reversible cellular Automaton should behave exactly that way! And from this common rigorous rule, nevertheless, there is EXACTLY ONE exception. And let's describe it. Fig. 4. What is the movement of the reversible cellular automaton (and our soliton, among others) with the beginning at white and blue plate. So, we construct our cellular soliton. It is constructed extremely easy … 1). The laws of transition are as follows: if any mask THAT HAS THE SYMMETRIES OF THE SQUARE has at least one point C -the law of transition is (II). If not -the law of transition is (I).
Choose 10 -from the head! -such masks, the rank doesn`t exceed 4. 2). We choose the size of our square area as 70x70 -and we surely CLOSE IT IN THE TORUS! 3). Now it is left to specify the question of initial conditions… We choose the following variant: all the cells are in the state A and only a few -random -are in the state B (Fig. 4). Then in the next step the blue ones will go into red, and the cells A will remain cells A... And this means that the automaton already at the first step will pass into its transliteration, and immediately will go back in time. And every state of the automaton at point of time t will be a transliteration of the automaton in the point of time -(t-1). Some time these two movements -forward and backwill close up.
We call the moment when it happens as a Point of «mirror» or a Point of half-cycle (T 1/2 ). We choose and write down for further use 5 sets of 8 random cells B. («Random» -in the sense that the coordinates of cells are given by the library function rand (). X = rand ()% 70. Y = rand ()% 70. We want to recall that. All the other cells are cells A).
So in total we have 5 (the number of sets of initial points) * 10 (the number of masks) = 50 different cellular automata. Let`s start them up… 3 First meeting with the Cellular Soliton Initially, just the first 10 -20 steps, there is an "explosion". Rapid "erratic" movement, which then ... at some point of time "calm down" somehow and the automaton begins to "work with a period 6". (Period -is a number of steps through which our automaton comes in approximately the same condition. By the way, this gives us an opportunity to shoot cartoons.) Automaton distributes somehow its cells in 6 phases, and itself creates a very interesting structure -so-called River System with Different Banks.
In six steps there is no change at the Bank. And the Rivers, by contrast, are constantly twisting a little and changing their riverbeds. And just in place of these "twisting places" they are permanently as if "washing out and modifying the" the Bank at this place.
For simplicity let`s introduce the following definitions. We call the River that encircles the whole torus -SuperRiver. One that doesn`t encircle will be called just the River. (There are also very distinctive NullRivers, to which we will come later!) In the Bank the most cells СВСВАА vary with a period 6 and with the appropriate phase shift. In the Rivers the cells vary with the period 3: СВАСВА... So...There has been an "explosion"! The Rivers were formed. And they were becoming stronger. That river which is bigger "eats" that river, which is smaller. To do this, the Rivers are sure to have to be in the appropriate phases. But fortunately the Rivers pulse, and at the time of passage through the zero they shift their phase. When the phases go well -the fusion takes place. They merge, merge...And our last SuperRiver formes. At some point in time it also pulses, but... At this point, when passing through zero, it will turn out that on some of the phases all points C disappeared, and this means that the Automaton has reached a Point of mirror. We`ll say that, in this case, the automaton has closed. (Or has returned). We have described a "canonical case". But if in the table in figure 4 there is a bold circle, it means that the automaton closed in a different way. Without using the last SuperRiver.
But, in general, all cases are of extremely peculiarity -we will talk more about it! (If in the table there is an icon <>, and above it there is a figure k -this means that the last SuperRiver while pulsing "didn`t close gently, through zero", but broke during the transition into k clusters which worked for some time before the final closure. If there are two icons <> -this means that once two SuperRivers were formed, which later for a long time tried to figure out which one of them is "more important". And so on ...) We put our mask on a chess board and count how many cells of the mask fell on black squares, and how many of them fell on the white. It is easy to see that only masks, in which that number is approximately equal, work well. (The masks made up exclusively of black or exclusively of white cells -are "not working" at all! It is tested!) Approximately the same words are referred to the initial points B, although here it is no longer so critical.
(... By the way, here you can ask a million questions: "What will happen if we take not a square, but a rectangle", "what will happen if we take the mask with the symmetries of a rectangle, not a square", "what will be ... "? But to begin answering them and giving schedules -would mean to go the "wrong way". Our urgent task is to drive the behemoth in a matchbox. And so we shouldn`t distract.
We`ll give just two figures. Let us ask ourselves: how the "median" of the value Т 1/2 -a number that for 50% of the tests under these conditions will be less than his, and for 50% it will be bigger -depends on the size of the site? The answer is: for the mask of the square 9x9, on the segment with sizes 30 -80, with an increase in the size of the square for 10, it increases approximately in 2.2 times. The dependence is approximately linear.
Let`s wonder what will be with an increase of the number of our original cells B. The answer is: with an increase of the number of the cells, the situation in the ideological sense, becomes much simpler! The question "will the soliton return in a small number of steps or no" depends on: whether will be the SuperRiver formed in the first 100 -200 steps? If it is formed -then everything is fine! It will surely -with a guarantee! -"clean" everything, and the soliton will close in less than ~ 20000 steps. If the answer is "no" -then the time of returning will be much, much more. Incidentally, this fact allows us to find easily absolutely magnificent examples of the "returns"! We can take a very large, for a home computer, the area size: 120 x 120. VERY big for it the value of number of initial points: 30. And then, take and sort out random Automata. Each time we should literally look through the first few hundred steps, if the SuperRiver was formed or not? If -"no", we should go to the next set. But if it is suddenly formed, then the soliton will surely close within the value of ~ 100,000 steps.
For our "good masks" the value of the number of points for which in half cases the SuperRiver is formed, and in half -isn`t formed, is ~ 16. That is, if in the table there were the median of appropriate half-cycles, approximately to the value of 16, there would stand a number ~ 15,000, and after -it would jump very abruptly.
We believe that for the first acquaintance it is enough! These examples will be enough for us till the end of the article. Briefly we`ll give the last thing that is shown on the figure 5. Green highlighted solitons are those, which will be described in Chapter 6. In the upper right corner the characteristic numbers of these solitons are shown. See Chapter 7).

How to examine DCS. Filters
Just look at the soliton -is not correct. We should look at it through the filter. And our goal is to make such filters that would work "in two directions": in the direct (in the direction of time), and vice versa. And to portray it between two successive frames. For this we take six consecutive frames of motion of our soliton, as it is shown in the figure 4. The frames 0, 1. 2, 3, 4, 5. Just in the middle -between the frames 2 and 3 -and we shall represent a "filter". Our picture of all these six frames.
1). Our first filter -the filter of the number of cells А in our six frames t-3, t-2, t-1, t, t+1, t+2 which have the coordinates x and y. It may have 7 values, from 0 to 6. Let`s note an obvious formula (see figure 4).
for any t, x, y. We know that during steady motion of a soliton, the cells change, in general, СВСВАА or СВАСВА, with different phases, and this means that in most cells for a variety of frames А F = 2. (By the way, these two cases, sort out all the possibilities for exactly two cells A in six frames). So let's paint the first case (CBCBAA; Bank cells) gray, and the second one (СВАСВА; River cells) green. The other colors are shown in figure. However, except for the first few frames, the number of А F could basically be equal only to 1 or 3. Well, maybe (rarely), to zero or to four, when some river, oscillating, passes through 0.
At once we should note very beautiful objects, which in the original, not filtered image were not visible. We called them NullRivers. Typically, they enter in gearing with the ordinary Rivers or with the others "Null". But we can easily find the pure one. In the figure 7 there are two "pure" at once.
NullRiver -is, indeed, the river with the length of zero. This is a single point of another color standing on the bank at a sufficient distance from the nearest River. And although the location of the nearest cells around it is different, the NullRiver on its own looks at our filter wonderfully without a single defect. (We should note that its period is equal to 12, and that their position is not directly connected IN NO WAY with the initial cells B! They are formed in absolutely random places). We'll talk about this in Chapter 9.

2). Our second filter
This filter puts the color on the boundaries (boundary) between the cells and fixes the "changes" at the borders. It depends on the same index "i", which can have three values: 0, 1, 2. ("j" has two indexes, and shows what kind of boundary is it -the horizontal or the vertical; x and y -as before, are the coordinates of the cell).

A t x y A T t x y
The filters A F и B F 0 drawn between the frames 57 и 58 of a soliton 1(e). The value of half-cycle for it is 2526. So for the filter A F (the coloring of the cells themselves) drawn between the frames (2*2526-57, 2*2526-57+1) = (5001,5002) will look exactly the same and the coloring of the boundary B F 0 will change to «the opposite».
If the value of i = 0 -we take "the changes" between the frames t and t +1. If i = 1 "the changes" between the frames t-1 and t +2. If i = 2 'the changes' between the frames t-2, and t +3. What exactly do we mean by the word "changes"? Let`s imagine that on the all the boundaries of cells there are put numbers "0" or "1". "0" -if the boundary separates the cells of the same color and "1" -if their color is different. Let`s take, for example, two central frames t and t +1. If there was at the border (at time t) a "0" and became (at time t +1) a "0" or it was "1" and became "1" -let`s then set for this boundary B F 0 = 0, and we will not paint it. If there was a "0" and became "1", let`s then set for this boundary a number B F 0 = -1 and paint the border in white. If there was a "1" and became "0", let`s then set for this boundary a number B F 0 = 1, and paint it in black. Let`s note the obvious formula: If we look at the figure 7 and wonder how will our filters look like for the same frames, but after a half-cycle (see fig. 4), the coloring of the cells will not change, and the coloring of the boundaries will change to the opposite: the black boundaries will become white, and the white will become black.
For the Filter B F i two simple formulae which will be true for any reversible CA can be proved.
But, in general, the filter B F i disappoints. Even seemingly true statement about the closure of "black and white" lines in fact turns out not to be true. For the mask (e) the violations also exist, although they are rare. For the mask (j) -they are ubiquitous. It seems that the only thing for which they can be useful -is to determine the "direction of movement" of the Rivers and SuperRivers in space, what we are going to talk about in the next chapter.

3). Let`s introduce a third filter
it also depends on the three indices, as a filter B F , but paints in turn, not boundaries, but the cells, as a filter A F . (C-coinsedance; the filter of "matches A").
Let`s suppose that we are building this filter for a zero index, that is, to the frames t and t +1. We`ll mark those cells that are in a condition A on these frames as a 1, and those which are not as a 0. And we`ll choose the function "Exclusive OR" between them. The result is our filter C F 0 . The filters C F 1 and C F 2 are constructed similarly. And if it is equal to 1, we will denote this filter with a small black cross in the center of the already colored by the A F filter cell. For it, as for the filter A F , obviously, the same formula is true: Let`s examine how look all three filters for the half of the separate SuperRiver. See fig. 8. They look very typical! Black and white curve of the filter B F 0 alternately passes from one side to another and then disappears for one frame. The polylines of the filter B F 1 border the whole river. Near the river, touching it, there are only "red" and "blue" cells. (We`ll recall that the "blue" cell is one cells A in six frames, "red" -cell is three). There is no others! The filter С F 1 also looks very beautiful. It fills alternately one half of the torus relatively to the SuperRiver, then a second half, then both of them...and then all repeats! Obviously, this is the way a pulsating River looks. During pulsation the phase changes and the rhythm is got out. Let`s suppose that our SuperRiver is ready to pulsate. How with the help of the filters, looking only for a few frames, establish in which half of the torus it is going to happen? Curiously, this seems to need all three of our filters at once. It is easy to see that for one half of the SuperRiver these filters, which are -A F , В F 1 , C F 1 -can occur only in the configuration shown in the figure 9. This makes it possible to determine the direction of its movement. In our case, the arrows will point to the location of pulsation. When the SuperRiver passes the pulsation, the arrows will point to the contrary -"to the outside".

C t x y
Let`s wonder if it is possible to guess from the same number of frames of the SuperRiver, in which half (before the half-cycle or after) we are?
The answer is no! The SuperRivers outwardly look exactly the same! And what can we say? Very little! We can only say that exactly in this half of the torus, sooner or later, "something interesting" will happen! The River will either close quietly, or vice versa, will produce the new Rivers ... then new ones ... then "anti-explosions", the Start Point, an "explosion" ... and so on. But we can not say what exactly! The author has an intuitive assumption that it is fundamentally impossible.
Actually, the filter В F 1 + C F 1 from the most elementary steps impress very much: with the closure of the filter B F , and with the fact that, В F 1 mainly goes in the boundaries of the areas with C F 1 =0 и C F 1 =1...but only for masks (a), (b) , (c)! For the mask (e) and especially (j) there are different disorders. But we do know that all these masks return equally well, and this means that in our consideration these filters are useless! 6 Key DCS graphics. "Local" time reversal Obviously, if there are numbers N C (0), N B (0) and N A (0) (initial number of cells C, B, A in the automaton), and the graph N C (t) (number of cells C after t steps), we can immediately build the other graphics: N B (t), N A (t). Therefore, we will consider only the graph N C (t).
Later ... Approximately by the tenth run our River Systems with Different Banks is forming well. And from the assumption that any Bank has approximately equal number of the cells in 2 colors follows that this graph (N C (t)) is roughly continuous with a period of three. Therefore, we will consider the N C (t) with three phases. The green line will denote the phase 0 -N C (t/3). The blue one -phase 1, N C (t/3+1). And the red one -phase 2, N C (t/3+2). (t/3 is the integral part of the division of t to 3). Fig. 10. Twelve of the fifty graphs N C (t). All of them returned in 100,000 steps with the exception of case 5(g). Yellow circles mark the moments of local time reversal. A number inside them is the number of cells C at this time. (That is, at the very end of the all cases, except for the 5(g), there is a "yellow circle" with the number 0. This is the moment of real time reversal).
The figure 10 shows the 15 graphs N C (t) of the movement of our solitons. All the cases returned, except for the 5(g). For the cases which returned it is easy to understand how will look our graphs after the point of half-cycle. Let`s take as an example the case 1(c). There the green phase turned its N C into zero, and hence it will reflect strictly symmetrically relatively to the point of half-cycle. The remaining two phases will also reflect strictly, but the red line will go into blue and blue into red.
Let`s consider the points marked with yellow circles on the graphs. It is evident that this is "almost" time reversal, or reversal in a large area of our torus. We will call them local time reversal. (The number in the circle itself is the number of N C at this point). There are many mysterious things here. Let`s take, for example, the case 1(e). There is present the pulsing SuperRiver and somewhere near there is a large cluster of Rivers and NullRivers. And ... at the very moment of pulsation, when the number of N C in the area of SuperRiver reduces to one (!)... at the same time the cluster (for some reason?) reduces its value of N C to a record of two (!). The Automaton has passed the point of "local reversal" and "almost" went back in time. The number of cells C increases rapidly to several hundred! ..
But having come to its virtual "start", it would no longer be like before. In case 1(e) the automaton has worked for some time and closed successfully. The case 4(j) is interesting. The automaton came to a virtual start and shut down literally immediately. In the case 5(g) the automaton goes back in time almost to the time t = 0 ... and there is a "disaster"! The automaton by the explosion is divided into a very large amount of NullRivers and 100,000 steps were not enough for it to close.

The Main Conservation Law of the DCS
Let`s go to the positive moments… Let`s ask ourselves: if it is «soliton», so where is the «infinite number» of the conservation laws?
We`ll find them easily!.. Thus, we`ll present our Main Conservation Laws (MCL) of our soliton: For any soliton, for any x and y where l -is whole number equal to the constanta! Let's discuss it in details...If we take A F at the moment of time t = 0, it is easy to see that it is even in the all cells, since the frames -3, -2, -1 are the transliterations of the frames 0, 1, 2. For the same reason, the number A F at the moment of time t =T 1/2 is even too. Let`s subtract 2 from them, divide in half and add. And to the whole number, that we got as a result, we add a fullfledged A F -2 throughout the motion from t =1 to t = T 1/2 -1.
As a result, in all cells we`ll obtain the same number multiple of 4. We`ll divide it by 4 and call as a characteristic number of this soliton. By analogy we`ll denote it with the number l. (It`s they which are given in the table in the figure 5, in the upper right corner).
The number of conservation laws is not "infinitely large" as in the usual soliton, but it is also very considerable. Their amount is as big as the amount of the cells in our area, namely 70*70 = 4900! The law mentioned above is tested… maybe for 100 very different solitons! And the "right" ones (described in the article), and the "wrong" ones. (Our conditions may be changed a little -for instance, no to close one side in the torus, or to go to the site 70 x 71 -and the soliton will return again. Sometimes. So: the Main Conservation Law is still immutable!) These Laws are as if cementing our narrative. It becomes clear that we are talking about the Phenomenon, which is uniform in spirit. The author doesn`t know anything about the nature of these Laws and the meaning of l, and he considers it inappropriate to talk on this topic.
The behavior of the function F from t is very interesting: .
First, it dramatically swells out on X and Y, and then, as if cools. But what is interesting is that almost at the all points x, y the function F on the whole movement -stays even. And always only in some points -it is odd. (Then it will be divisible by 4, and besides, it will be the same at all the points after the addition of the last "half". At the last step only a few points are added with A F -2 = -2 (can be zero points). And a few points with A F -2 = 2 (maybe, too, zero points). But, apparently, both of them can not be zero.)

The main integral of the DCS
The conservation laws lead to the concept of the Main Integral of the DCS -function -S. The values of all the sums values (A F -2) for all x and y from t 0 to t. (And if t 0 =0 it is necessary to take the half of this sum. If t=T 1/2 then it`s necessary to do the same). ..
.. Fig.11. The behavior of the function S(0,t)=S(t) for eight solitons. Only one of them -the variant 5(g) didn`t return.
The figure 11 shows the function S(t)= S(0.t) for eight solitons. In this figure it is perfectly clear that our "local time reversal" (some of which are outlined with rectangle) neither are a false artifact nor the "aberration of view". It is passage by the function S «not its l». Well, maybe "its", but "not at this time '(soliton 1(h)). (The case 2(c) is special!) It is seen that the solitons prefer to do it at speed, and it seems so, that with the same derivative. Again, it seems that the variant 5(g) did not succeed in doing it.
It`s interesting that function S(t) may come both from above and below to the Point of half-cycle. And now, the reason why we named our function S(t) -the Main Integral...

The main «symmetry» of the DCS
And our final observation...It concerns the connection of the function S and N C . We called it the main "symmetry" of the DCS.
(The word is in quotes because it is not quite correct, it is"fuzzy". But despite this, it is very, very "powerful" and therefore -the Main one). We are not denying the pleasure to tell you how we came to it... When looking for a long time all over the figure 11, a strange feeling of "deja vu" appears. Have we seen it somewhere already?..Well, of course! On the figure 10. Moreover, we have seen exactly these graphs. But with a little explanation...
Let's go to the other end, and start from the Point of half-cycle. And at once we`ll write down our formula for symmetry. And now we`ll leisurely explain everything shown on the figure 12. Let`s arrange the graphics. To arrange the graphics N C and S on t it is just needed to reflect all of them relative to the vertical line. So let's do it. (We are not concerned that we may get the wrong colors!)

M t M kS t M k M
We should select k for the graph S. Let`s suppose that we have guessed and have chosen k = 0.25. (If we have not guessed, it is easy to adjust it to about exact). And we can see at once that the graph S(t) suspiciously exactly resembles...the "median" graph of our graphics N C ! The "median" graph -M(t) in the formula mentioned above -is the average graph of three; it is easy to see that it is also always about continuous.
But the second half of the black graph S, for some reason does not coincide with the median. It would be good to reflect it relative to the horizontal line at the point A. And then we see that at this very moment -at the point A -the colors on the median changed. It was green, and became blue! And we understand that the solution is found ... It is just necessary to introduce a signature -sgn(M) -for our Main Integral and change it to the opposite, when the median color changes. It's very simple. We check the same principle at the soliton 4(j). Here we have not only one, but 4 changes of color. We see that here everything also coincides.
Relative to the value M 0 ... It may be equal to the number of white cells in the picture of half-cycle, may be equal to the number of blue ones. Compare the graphics 1(b) and 1(c) from the figure 10. In the case of soliton 1(b) after a point of half-cycle the median graph will become red (phase = 2), and in the case 1(c) -blue (phase = 1). The initial signature depends on how the Main Integral function comes to the point of half-cycle: from the the top or from the bottom. (Fig. 11).
But if everything is chosen correctly, then on the figure 12 we see an "almost perfect" coincidence, the functions M(t) and S plus-minus from t. Then the graphics will slowly begin to diverge, and it is difficult to say what is the reason for this? The fact that we have chosen a little bit wrong k, or...The fact that there is an obvious drawback of the scheme. Every phase is average at the moment, we can determine the average once every three frames, and the total amount should be taken every frame! And we can show that it is important! The amount graph, quite a bit, but jumps at the moment of color change. And over time, it will "jump" far enough. (See fig. 13). But again you can "put" one graph to another, and continue ... Again, at first everything is perfect ... Curiously, that k does not change throughout the movement! This new constant is unique for each soliton. We have shown them for our seven solitons in the figure 11.