The new important property of wide class PDE was found solely by K. A. Volosov. We make an arbitrary replacement of variables. In the case of two independent variables

A new method of construction of exact solutions for partial differential equations (PDE) is proposed in this article. The classical authors in mathematics used change of variables for classification of linear PDE. However, they did not notice an important property of broad class of PDE, which was discovered in Volosov's articles [

Let's make an arbitrary replacement of variables:

We note that

An inverse transformation exists, at least locally:

The derivatives of the old independent variables on the new variables are determind as follows:

As

Taking into consideration on (

System (

The implicit linear algebraic equationsystem (

Equation (

We can express any three derivatives

We can substitute it to (

We obtained

Matrix

Eigen pair can be discovered easily. We are not going to discuss here their interesting properties.

At the second stage, consider the new first-order system (

It is well known that the solvability of a system of this type is verified by calculating the second mixed derivatives of the functions

(1) one has the new identity

(2) Two solvability conditions (

If some free functions

This property (Theorems

The specific ways of satisfying the conditions of (

Let's consider the Zel'dovich equation, which is well known in combustion theory [

Suppose that

In the articles [

Let one

We can try to satisfy this (

It turns out that in a number of interesting cases, all the four equations can be solved. Moreover, the function

Equation (

System (

The exact solutions of the PDE are already constructed, since relations (

Let's choose the function

Then, we have the following system of two equations for

It turns out that, if we can do the variables replacement

The authors believe that the solution (

We can came back to the variables

Let's consider (

We have a system of two equations for

Suppose that (

Let's express from (

Consider the case

Suppose

This is a new real family of solution to (

Consider the Fitz-Hugh-Nagumo-Semenov semilinear parabolic partial differential equation, which is well known in biophysics theory [

In this case, equation analogous (

Suppose

The denominator in the system of ODE (

Then the exact solution of (

Suppose that (

After that, we came back to the variable

Let's substitute this expression in the first parity (

In the three-dimensional case, consider the semilinear parabolic and the change of variables:

Let's supplement the relations written above by the equalities of mixed derivatives

The nonlinear algebraic system of seven equations in the variables

Suppose that

Then, the exact solution of (

The function

The twice differentiable functions

If

Let's consider the modification equation Fisher-Kolmogorov-Petrovsky-Piskunov

We can choose

Systems (

The exact solutions of the PDE are already constructed, since relations (

Let's assume

Denominator

The exact solution of (

Exact solution equation of (

Exact solution of system (

The following new facts complementing PDE theory were discovered in the study of Mr. K. A. Volosov.

The system of four first-order equations (

There is a new identity which allows to record monomial factor (

Monomial factor (

Using this approach it is possible to construct new families of exact solutions in the parametric form, the examples of which were presented in this article. These solutions cannot be constructed using the classical method of batch properties analysis.

It is useful to study the Eigen pair's properties for matrix

Below you can find the extract from the comments to the article of Mr. M.V.Karasev, Doctor of Physics and Mathematics, Professor, Head of Chair of Applied Mathematics of Moscow State Institute of Electronics and Mathematics, Laureate of State award of Russian Federation.

The article contains several interesting solutions and whole classes of solutions for nonlinear differential equations with partial secondary derivatives (PDE) important in applications. Sometimes in the theory of PDE the approach is used when by making certain transformation (e.g. by replacement of variables) which is converted in order to bring it to another equation whose solution is already known. Thus, the known solution generates nontrivial solution of initial equation. In Mr. K.A. Volosov's approach it is suggested a priori not fixing the type of replacement of variables, leaving that arbitrary on the first stage. The system derived after replacement of variables contains both required vector-function (sought after solution and its derivatives) and unknown coordinate transformation (Jacobi's matrix of variables replacements). The number of equations in the system is always less than the number of variables which opens possibilities for construction of solution by introduction of different interconnections between the components of sought after vector-function. Limitation for the choice of connections requires consistency of the equation for coordinate replacement function co-ordinate replacement function must be consistent. A condition of consistency is mandatory for the remaining independent components of sought after vector-function.

For example, in classic case, for PDE second order with two independent variables, the sought after vector-function has three components; coordinate replacement contains two more functions; and all of them together subordinate to four differential equations of first-order. It was found that all the components of Jacobi

This algorithm is not tied to any group or symmetry attributes. The proposed transformation of nonlinear differential multi variable equations has common nature with the mechanism of integration. The coefficients and type of nonlinearity in the equation solved are not specified, but remain general functions.

This work opens very interesting line in many areas of mathematics and its applications dealing with nonlinear equations with higher partial derivatives.

The author is grateful to V. P. Maslov, M. V. Karasev, S. Yu. Dobrokhotov, V. G. Danilov, A. S. Bratus for attention and advice.