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Waves on water long since attracted to themselves a of researchers. It is connected with that they represent widely known phenomenon in the nature and, besides, accompany movement of vessels on water.

In practice it is difficultly to create a monochromatic wave, and usually deal with Zug (package) of waves in which each wave extends with the speed, and the speed of distribution of a is characterized by group speed

Let's enter linear space of h of net functions, on a grid with values in knots of a grid of yi=yh (xi). It that conditions of frequency of y0=yN are satisfied. Besides, formally we believe yi+N=yi for i 

Using special approach, it is possible to that the principle of superposition of decisions for Kortevega Fris's equation is not carried out and therefore this equation is nonlinear and describes nonlinear waves.

in which and, k and  — constants, at  = ±k is the solution of the equation (. In this decision and — amplitude, k — wave number, and  — frequency. The decision represents the wave postponed in the environment with a phase speed

The real work is devoted to research of the equation of Kortevega – de Friza. The extensive literary review on a research subject is carried out. Various differential schemes for KDF equation are studied. The practical account with use obvious five dot peddling schemes is executed

However if to think, such to a lonely wave of Russell becomes. The matter is that owing to the specificity this opening long time was considered as quite fact. Really, at that time the physical world seemed linear and the principle of superposition was considered as one of the fundamental principles of the majority of physical theories. Therefore none of researchers did not give to opening of an wave on water of serious value.

Actually this equation is to a the confidant, as at his conclusion of an small parameters ( . If ­ influence of these parameters, directing them to zero, we will receive one of parts of the decision of D'Alamyober.

After application several times of integration in parts and fourth integrals are reduced. The second and third the disappear because of boundary conditions. Thus from the first integral we receive:

where g — the acceleration of gravity, and a

Lonely waves which were opened by, and really behave as a particle. The wave does not pass through small at their. When lonely waves adjoin, the big wave is slowed down and decreases, and the wave which was small, on the contrary, is accelerated and grows up. And when the small wave grows to big, and big decreases to the sizes small, solitons are divided also the bigger leaves forward. Thus, solitons behave as tennis balls.

Coming back to waves on water, we will notice that they can be analyzed using the well-known equations of hydrodynamics of which it is known that they are not linear. Therefore also waves on water generally are nonlinear. Only in a case of small amplitudes these waves can by the linear.