The international scientific magazine "Chaos, Solitons and Fractals", the second most important magazine of mathematical physics with the Q1 quartile, published an article (external link) by Nikolai Kudryashov, Professor of the National Research University of MEPhI, "Solitons of the hierarchy of the complex modified Korteweg–de Vries equation".

The study of solitons (structurally stable solitary waves) is one of the most difficult mathematical problems with great prospects and a fascinating history.

"In 1965, American mathematicians Martin Kruskal and Norman Zabuski conducted a computational experiment on the interaction of solitary waves, which they called "solitons," says Nikolai Kudryashov. – And we have received a completely paradoxical result – the actions of solitons are very similar to the behavior of particles or even to the actions of people: a large solitary wave that is moving at a higher speed catches up with a smaller solitary wave moving at a lower speed, they interact, then scatter without changing shape and speed. The only thing is that they settle for a while, as if "talking to each other" and run away. It turned out that this phenomenon occurs in many areas of physics where scientists try to take into account nonlinearity and dispersion (wave scattering) – in plasma physics, solid state physics, optics, superconductivity. The simplest example is that it can even be used to describe some waves on the water, for example, a tsunami."

But in the process of solitons discovery not only mathematics, but also geography and poetry take part.

In 1895, Dutch mathematician, mechanic and astronomer Diederik Korteweg and his graduate student Gustav de Vries first mathematically explained the phenomenon noted in 1844 in an article by British scientist and shipbuilder John Scott Russell. Russell made an experimental discovery in 1834, and it happened almost by accident when he was testing a barge on the Forth and Clyde Canal, which cuts Scotland into two parts. The essence of the experiment described two horses pulled a barge along the canal, and Russell, sitting on the third, followed them along the shore. Suddenly, something scared a couple of horses and they bolted, the rope broke, and the ship stopped as a small wave from the bow went forward in front of it. Russell, who had extraordinary observation, became interested in this phenomenon, and drove along the shore behind the wave to monitor its changes. Over the course of ten years, he repeated this experiment many times intentionally, that is, he observed and studied the movement of a solitary wave on the surface of the water. He published his conclusions in 1844 in the "Report on Waves", but he was criticized and for a long time this work did not arouse interest. The founder of modern hydrodynamics, George Gabriel Stokes, especially "tried" calling Russell's scientific research "nonsense".

It was only 50 years later that Korteweg drew attention to Russell's report and together with de Vries they derived an equation describing the propagation of a structurally stable solitary wave, that is, a soliton, in a nonlinear medium.

And now about poetry. In 1841, that is, at about the same time when Russell was conducting his observations in Scotland, Mikhail Lermontov wrote about it in Russia!  Here is a poem that he left in Countess Rostopchina's album:

I believe: under one star

You and I were born;

We walked along the same road,

We were deceived by the same dreams.

But well! – from a noble purpose

Torn away by a storm of passions,

I forgot in the fruitless struggle

The devotion of my youth.

Anticipating eternal separation,

I'm afraid to give my heart free rein;

I'm afraid of the treacherous sound

To entrust a dream in vain...

So two waves rush together

A random, free couple

In the desert, the sea is blue:

They are driven together by the south wind;

But they will be separated somewhere

The cliffs have a stone chest...

and are full of the usual cold,

They carry various shores,

Without regret and love,

His murmur is sweet and languid,

Its stormy noise, its borrowed brilliance

And their eternal caresses.

This is nothing more than a poetic description of a nested soliton, which has the same properties as the Korteweg–de Vries soliton. Whether the poet invented it or noticed it at sea, we do not know. By the way, the Lermontov family comes from the Scottish bard of the XIII century Thomas Lermont. "This is how you start studying family portraits and, perhaps, you start believing in the transmigration of souls," as Sherlock Holmes said. Whose prototype was a surgeon and professor at the University of Edinburgh Joseph Bell, was also a Scot. Apparently, there is some kind of mathematical mystery in the existence of Scotland itself...

But let's return to the academic class. In 1945, the scientific community of Europe celebrated the 100th anniversary of Korteweg's birth. Interestingly, the Korteweg–de Vries equation was not even included in the list of outstanding works of the scientist at that time.

Now fast forward to the USA, to the Los Alamos Laboratory, where in the forties there was intensive work on the creation of atomic weapons and ... computing machines, the first of which appeared in the early 1950s. One of the active participants in the creation of atomic weapons in the USA was an outstanding Italian physicist, Nobel Prize winner Enrico Fermi. He proposed solving not only "explosive tasks" on the first computer, but also purely scientific ones. At that time, Fermi was trying to figure out a mathematical model of the infinite velocity of heat propagation, which follows from the equation of thermal conductivity, but contradicts Einstein's theory of relativity.

Fermi suggested that if, when transmitting a perturbation in a discrete chain of atoms, non-linearity is taken into account, this will correct the mathematical model and lead to a finite rate of heat propagation in a solid. He assigned this task to two of his colleagues, engineer John Paste and mathematician Steve Ulam, to solve it on a computer, since there were no other ways to do this.

However, after performing a computational experiment, the researchers did not get the expected result. Rather, on the contrary, they got what they could not explain: when calculating perturbations in a chain of atoms under periodic boundary conditions, the initial state returns. It was unexpected and incomprehensible, and remained in the history of science as the Fermi–Pasta–Ulama paradox.

Martin Kruskal and Norman Zabuski were lucky enough to solve this paradox in 1965. Kruskal moved from a discrete chain of atoms to a continuous mathematical model, assuming that the number of atoms in the chain tends to infinity, and aiming the distance between them to zero. In fact, he came to the string, taking into account the nonlinear interaction in it during the propagation of perturbations. And then the unexpected happened: he got the Korteweg–de Vries equation. At first he thought he had obtained a new equation, but after consulting with one of the professors at the Department of Hydrodynamics at Princeton University, he learned that the equation he had obtained had already been known since 1895 to describe waves on water. This is how the Korteweg–de Vries equation returned to scientific circulation, it finally came in handy and opened up a whole "bouquet" of new directions – in physics, hydrodynamics, shipbuilding, oceanology, etc.

"The mathematical model for describing solitons really excited the scientific community in the late 60s of the twentieth century, for its solution. The Korteweg-de Vries equation was written in the form of linear equations, which led to a new method for solving Cauchy problems for nonlinear partial differential equations," Kudryashov continues. – It was a remarkable discovery, after which the "soliton fever" arose, and other equations with solitons were searched for. And indeed, it turned out that solitons occur in different models – there are group solitons, topological, magnetosonic, gravitational and others."

The phenomena described by these equations can be predicted, but it is very difficult to solve such problems. In the first half of the 1970s, Japanese mathematician Rego Hirota came up with an elegant method for finding soliton solutions - now it is called the "direct Hirota method". It was used for equations of the second, third and fourth orders.  Kudryashov applied the Hirota method for equations of higher orders – fifth, seventh, etc. In particular, for the complex modified Korteweg–de Vries equation.

"As far as I know, this has not been done in the world yet. But it wasn't that simple. The first difficulty was to figure out how to write a high–order equation using the Hirota operator, and the second was how to split one equation into a system (that is, to get two from one equation) so that Hirota's ideas could be used. The work was quite difficult, the calculations were complex and cumbersome, but as a result, it was possible to find two–soliton solutions of equations of the fifth and seventh orders for the hierarchy of the complex modified Korteweg–de Vries equation," the scientist concludes.

In prof. Kudryashovs’ work  one- and two-soliton solutions of equations of the fifth and seventh orders were found and a hypothesis was expressed about the form of a multi-soliton solution for the remaining members of the hierarchy – the ninth, eleventh and other orders. The considered equations can be useful in the development of mathematical models for transmitting data over long distances without interference, including in nonlinear optics – to describe pulses in optical media.

Congratulations to Nikolai Alekseevich and wish him further success!