Decaying and degenerate surfaces of the second order. Decaying and degenerate surfaces of the second order Decaying equations method for their solution

Today we will continue to work with you on the topic: whole equations

We have to consolidate the skills of solving equations with a degree higher than the second; learn about the three main classes of entire equations, master ways to solve them

On the back of the board, two students have already prepared solution # 273 and are ready to answer students' questions

Guys, I propose to recall a little the theoretical information that we learned in the previous lesson. I ask you to answer the questions

Which one-variable equation is called an integer? Give examples

How do you find the degree of an entire equation?

What form can the equation of the first degree be reduced to

What will be the solution to such an equation

What form can an equation of the second degree be reduced to?

How to solve such an equation?

How many roots will it have?

What form can the equation of the third degree be reduced to?

Equation of the fourth degree?

How many roots can they have?

Today, guys, we will learn more about entire equations: we will study ways to solve 3 main classes of equations:

1) Biquadratic equations

These are equations of the form
, where x is a variable, a, b, c are some numbers and a ≠ 0.

2) Decaying equations, which are reduced to the form A (x) * B (x) = 0, where A (x) and B (x) are polynomials with respect to X.

You have already partially solved the decaying equations in the previous lesson.

3) Equations solved by changing the variable.

INSTRUCTIONS

Now each group will receive cards that describe the solution method in detail, you need to jointly analyze these equations and complete the tasks on this topic. In your group, check the answers with those of your comrades, find mistakes and come to a common answer.

After each group has worked out their equations, they will need to be explained to the other groups at the blackboard. Think about who you are delegating from the group.

WORK IN GROUPS

During group work, the teacher observes how the children reason about whether the teams have formed, whether the children have leaders.

Provides assistance if necessary. If a group coped with the task earlier than others, then the teacher still has the equations from this card of increased complexity in stock.

CARD PROTECTION

The teacher offers to decide, if the guys have not done this yet, who will defend the card at the blackboard.

The teacher can, during the work of leaders, correct their speech if they make mistakes.

So, guys, you listened to each other, the equations for your own solution are written on the board. Get down to business

Ur. Igr.

IIgr.

IIIgr.

You need to solve those equations that you do not have.

No. 276 (b, d), 278 (b, d), 283 (a)

So guys, today we studied the solution of new equations in groups. Do you think our work went well?

Have we reached our goal?

What hindered you in your work?

The teacher evaluates the most active children.

THANK YOU FOR THE LESSON!!!

In the near future, it is advisable to carry out an independent work containing the equations analyzed in this lesson.

"Solving equations of higher degrees" - What does it mean to solve an equation? Tasks of the first stage. WARM-UP (check for d / h). Solving equations of higher degrees. What kinds of equations are written on the board? Physical education. Stage II Independent work option 1 option 2. What is called the root of the equation? Scheme for solving the linear equation of the quadratic equation of the biquadratic equation.

"Methods for solving equations and inequalities" - Ancient Egypt. Cubic equations. Non-standard methods for solving equations and inequalities. The idea of ​​homogeneity. A graphical way to solve equations containing a module. Inequalities with the module. Solving equations for coefficients. The original inequality does not contain any solution. The sum of the square.

"Equations and Inequalities" - Substitution. Find the abscissa of the intersection point of the graphs of the functions. At what value of a is the number of roots of the equation. "The graphical method. It consists in the following: plotting the graphs of two functions in one coordinate system. Solutions of equations and inequalities." Find the smallest natural solution to the inequality.

Fractional Equations - Solve the resulting equation. A quadratic equation has 2 roots if …… Exclude the roots that are not included in the admissible values ​​of the fractions of the equation. … Your letter. High soul ". Algorithm for solving fractional rational equations. And remember - what is the main thing in a person? Fractional rational equations. How many roots does this equation have? 4. What is the name of this equation?

"Solving logarithmic equations" - If the equation contains logarithms with different bases, then first of all, you should reduce all the logarithms to one base using the transition formulas. Calculate the values ​​of the expression. Definition: To summarize the material on the properties of logarithms, logarithmic function; consider the main methods for solving logarithmic equations; develop oral skills.

"Methods for solving logarithmic equations" - Find. Solving logarithmic equations. What is called a logarithm. Systematize student knowledge. Creative work. Find the error. System of equations. Solving logarithmic equations by various methods. Option I Option II. The given function. Method for introducing a new variable. Compare. Methods for solving logarithmic equations.

There are 49 presentations in total

DOCUMENTS OF THE ACADEMY OF SCIENCES, 2008, volume 420, no. 6, p. 744-745

MATHEMATICAL PHYSICS

DECAYING SOLUTIONS OF THE VESELOV-NOVIKOV EQUATION

© 2008 Corresponding Member of the RAS I. A. Taimanov, S. P. Tsarev

Received February 14, 2008

Veselov-Novikov equation

u, = e3 u + E3 u + s E (vu) + zE (vu) = o, E V = E u,

where E = (Ex - ¿Ey), E = 1 (Ex + ¿Ey), is a two-dimensional generalization of the Korteweg-de Vries equation (KdV)

and, = 4 uhxx ​​+ viih,

into which it goes in the one-dimensional limit: V = u = u (x). Equation (1) defines deformations of the two-dimensional Schrödinger operator

specifies the transformation of the solution φ of the equation Hf = 0 into the solution b of the equation H b = 0, where

H = EE + u, u = u + 2 EE 1n w.

In the one-dimensional limit, the Moutard transformation is reduced to the well-known Darboux transformation.

Moutard transformation expands to system solutions transformation

Hf = 0, f (= (E3 + E3 + 3 VE + 3 V * E) f, ^^^

where Э V = Эи, ЭV * = Э и, which is invariant under transformation (extended Moutard transformation)

= ~ | ((f Eyyu-ef) dz- (f Eyu- ef) dz +

of the form H1 = HA + 5H, where A, B are differential operators. Such deformations preserve the "spectrum" of the operator H at the zero energy level, transforming the solutions of the equation

Hf = (EE + u) f = 0 (3)

according to (Er + A) φ = 0.

There is a method for constructing new solutions (u, φ) of equation (3) from old solutions (u, φ) of this equation, which is reduced to quadratures - the Moutard transformation. It consists in the following: let an operator H with potential u be given and a solution w of equation (3): Hw = 0. Then the formula

W | [(f Esh - w Ef) dz - (f Esh - w Ef) dz]

Institute of Mathematics. S.L. Sobolev, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Krasnoyarsk State Pedagogical University

+ [f E u - u E f + u E f - f E u +

2 2 "2 _ ~ _2 + 2 (E f Esh - Ef E w) -2 (E f Esh - Ef E w) +

3V (f Esh - w Ef) + 3 V * (w Ef - f Esh)] dt),

u ^ u + 2EE lnm, V ^ V + 2E21nm,

V * ^ u * + 2E21psh,

where w satisfies (4).

Veselov-Novikov equation (1) is

the compatibility condition for system (4) at V * = V.

When the solution w is real, the conditions u = u u

V * = V are preserved and the extended Moutard transform translates real solutions and

equation (1) into other real solutions and this equation.

All rational solitons of the KdV equation are obtained by iterating the Darboux transformation from the potential u = 0. Moreover, all the resulting potentials are singular.

In the two-dimensional case, a similar construction can lead to nonsingular and even rapidly decreasing potentials already after two iterations.

DECAYING SOLUTIONS TO THE EQUATION

walkie-talkies. Namely, let u0 = 0 and u0, ω2 are real solutions of system (4):

u, = Γ (z, z) + f (z, z), = i (z, z) + i (z, z), (5)

where / and π are holomorphic in r and satisfy the equations

fg = Yyyy "yag = yyyy"

Each of the functions uj u2 defines the (extended) Moutard transformation of the potential u = 0 and the corresponding solutions of system (4). Let's designate them as Mu and Ma. The resulting potentials we

we denote by u1 = Myu (u0), u2 = Myu (u0).

Let δ1 e My (ω2), i.e. b1 is obtained from ω2 by transforming M ω. Note that the Moutard transformation for φ depends on the constant of integration. We choose a constant such that b1 is a real function. The choice of a constant allows us to frequently control the nonsingularity of the iterated potential (we will use this in specific examples).

A simple check shows that b2 = - b1 f

e mu (yuh). The well-known lemma holds, which is true for an arbitrary potential u0.

Lemma 1. Let u12 = M01 (u2) and u21 = M02 (u2). Then u12 = u21.

For the case u0 = 0, we have Lemma 2. Let ω1 and ω2 have the form (5). Then the potential u = Mb (My (u0)), where u0 = 0 and δ1 e My (u2), is given by the formula

u = 2EE 1nI ((/ I - fya) +) ((f "I - fя") dr + + (GY - G I) dr) +1 (G "I - fя" "+ 2 (f" I " - GZ) + + GYa "" - G "" I + 2 (zi - zi ")) dz).

We note that even for stationary initial solutions ω1, ω2 of system (4), we can obtain a solution of the Veselov-Novikov equation with nontrivial dynamics in r.

Theorem 1. Let U (z, z) be the rational potential obtained by the double Moutard transformation from ω1 = iz2 - i ~ z, ffl2 = z2 + (1 +

I) z + ~ z + (1 - i) z. The potential U is nonsingular and decreases like r-3 for r ^ Solution of the Veselov-Novikov equation (1) with the initial data

U \ t = 0 = U becomes singular in a finite time and has a singularity of the form

(3 x4 + 4 x3 + 6 x2 y2 + 3 y4 + 4 y3 + 30 - 12 t)

Comment. The Veselov-Novikov equation is invariant under the transformation t ^ -t, z ^ -z. It is easy to see that the solution to this

equation with initial data U (z, z, 0) = U (-z, - z) is regular for all t> 0.

The rational potential (1) given in the work decreases as r-6 and gives a stationary nonsingular solution of the Veselov-Novikov equation. Choosing f (z) = a3z3 + a2z2 + a1z2 + a0 + 6a3t, g (z) = b3z3 + b2z2 + b1z2 + b0 + 6b3t, it is easy to obtain solutions of the Veselov-Novikov equation, decreasing at infinity, non-singular at t = 0 and having singularities at finite times t> t0.

Note that solutions of the Korteweg-de Vries equation with smooth, rapidly decreasing initial data remain nonsingular for t> 0 (see, for example,).

This work was carried out with partial financial support from the Russian Foundation for Basic Research (project codes 06-01-00094 for I.A.T. and 06-01-00814 for S.P.Ts.).

BIBLIOGRAPHY

1. Veseloe AP, Novikov SP // DAN. 1984. T. 279. No. 1. S. 20-24.

2. Dubrovin B A., Krichever I. M., Novikov SP. // DAN. 1976. T. 229. No. 1. S. 15-19.

It remains to consider the sets given by equations (35.21), (35.23), (35.30), (35.31), (35.32), (47.7), (47.22) and (35.20)

Definition 47.16.The surface of the second order is called decaying if it consists of two surfaces of the first order.

As an example, consider the surface given by the equation

The left-hand side of equality (35.21) can be factorized

(47.36)

Thus, a point lies on the surface given by equation (35.21) if and only if its coordinates satisfy one of the following equations or. And these are the equations of two planes, which, according to paragraph 36 (see paragraph 36.2, 10th row of the table), pass through the axis of the application OZ. Hence , equation (35.21) defines a disintegrating surface, or rather, two intersecting planes.

Task: Prove that if a surface is both cylindrical and conical at the same time, and also consists of more than one straight line, then it disintegrates, i.e. contains a certain plane.

Consider now the equation (35.30)

It can be decomposed into two linear equations and. Thus, if a point lies on the surface given by equation (35.30), then its coordinates must satisfy one of the following equations: and. And this, according to paragraph 36 (see p. 36.2 6th row of the table), is the equation of planes parallel to the plane. Thus, equation (35.30) defines two parallel planes and is also a disintegrating surface.

Note that any pair of planes and can be specified by the following second-order equation. Equations (35.21) and (35.30) are canonical equations of two planes, that is, their equations in a specially selected coordinate system, where they (these equations) have the simplest form.

The equation the same (35.31)

is generally equivalent to one linear equation y = 0 and represents one plane (according to paragraph 36, item 36.2, 12th row of the table, this equation defines a plane).

Note that any plane can be defined by the following second-order equation.

By analogy with equation (35.30) (at), it is sometimes said that equality (35.20) defines two merged parallel planes.

We now turn to degenerate cases.

1.Equation (35.20)

Note that a point M (x, y, z) belongs to the set given by equation (35.20) if and only if its first two coordinates x = y = 0 (and its third coordinate z can be anything). This means that equation (35.20) defines one straight line - the axis of the application OZ.

Note that the equation of any straight line (see paragraph 40, item 40.1, as well as paragraph 37, system (37.3)) can be defined by the following second-order equation. Equality (35.20) is canonical second-order equation for a straight line, i.e. its second-order equation in a specially selected coordinate system, where it (this equation) has the simplest one.

2. The equation (47.7)

Equation (47.7) can be satisfied by only one triple of numbers x = y = z = 0. Thus, equality (47.7) in space sets only one pointО (0; 0; 0) - origin of coordinates; coordinates of any other point in space cannot satisfy equality (47.7). Note also that a set consisting of one point can be specified by the following second-order equation:

3. Equation (35.23)

And this equation cannot be satisfied at all by the coordinates of any point in space, i.e. it defines an empty set... By analogy with equation (33.4)

(see item 47.5, definition 47.8), it is also called an imaginary elliptic cylinder.

4.Equation (35.32)

This equation also cannot be satisfied by the coordinates of any point in space, therefore it defines an empty set. By analogy with the similar equation (35.30), this "surface" is also called imaginary parallel planes.

5. The equation (47.22)

And this equation cannot be satisfied by the coordinates of any point in space, and, therefore, it defines an empty set... By analogy with the equality with equality (47.17) (see § 47.2), this set is also called an imaginary ellipsoid.

All cases have been reviewed.