CC BY
57
Серия «Математика»
2019. Т. 28. С. 53-68
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
ИЗВЕСТИЯ
Иркутского государственного ■университета
УДК 517.952,517.957 MSG 35F20,35F21,35F50
DOI https://doi.org/10.26516/1997-7670.2019.28.53
On Exact Multidimensional Solutions of a Nonlinear System of First Order Partial Differential Equations*
A. A. Kosov
Matrosov Institute for System Dynamics and Control Theory of SB RAS, Irkutsk, Russian Federation
E. I. Semenov
Matrosov Institute for System Dynamics and Control Theory of SB RAS, Irkutsk, Russian Federation
V. V. Tirskikh
Irkutsk State Transport University, Irkutsk, Russian Federation
Abstract. This study is concerned with a system of two nonlinear first order partial differential equations. The right-hand sides of the system contain the squares of the gradients of the unknown functions. Such type of Hamilton-Jacobi like equations are considered in mechanics and control theory. In the paper, we propose to search a solution in the form of an ansatz, the latter containing a quadratic dependence on the spatial variables and arbitrary functions of time. The use of this ansatz allows us to decompose the search of the solution's components depending on the spatial variables and time. In order to find the dependence on the spatial variables one needs to solve an algebraic system of some matrix and vector equations and of a scalar equation. A general solution of this system of equations is found in a parametric form. To find the time-dependent components of the solution of the original system, we are faced with a system of nonlinear differential equations. The existence of exact solutions of a certain kind for the original system is established. A number of examples of the constructed exact solutions, including periodic in time and anisotropic in the spatial variables ones, are given. The spatial structure of the solutions is analyzed revealing that it depends on the rank of the matrix of the quadratic form entering the solution.
Keywords: nonlinear system, Hamilton-Jacobi type equations, exact solutions.
* The research was partially supported by RFBR grant no. 19-08-00746
1. Introduction
The aim of the present work is to construct exact multidimensional solutions of the following nonlinear system of first order partial differential equations:
' ut = fh(t)\Vu\2 +ai(t)v +f(t), vt = f32(t)\Vv\2+a2(t)u + g(t).
Here, u = u(x,t), v = v(x,t), t € R+, x € Rra, n € N, n > 2, ut = du
—, V is the nabla operator; «¿(i), Pi(t), i = 1,2, f(t), g(t) are given functions of time t. First order partial differential equations of the kind (1.1) are considered in mechanics and control theory, in particular, when u = v, ai(t) = a2{t) = a(t), (5\(t) = fa(t) = f3(t), f(t) = g(t) the system (1.1) is reduced to the following multidimensional Hamilton-Jacobi type equation [7; 16]
ut =/3(t)\Vu\2+ a(t)u +f(t). (1.2)
When p(t) = -1/(2m), m € R, a(t) = f(t) = 0 from (1.2) we obtain the Hamilton-Jacobi equation for a free particle [1]:
ut + = 0, (1.3)
2m
where the function u(x, t) represents the action, m is a constant (the particle's mass). We note that the maximal local (pointwise) invariance group of Eq. (1.3) is also given and some of its exact solutions are provided in [1]. Therefore, the construction of exact multidimensional solutions of nonlinear systems of the form (1.1) is a quite interesting problem. In general, the construction of exact solutions of nonlinear differential systems is important in a number of other applications, for instance, in the theory of kinetic systems [8; 13; 14].
In this paper, we search exact multidimensional solutions of the system (1.1) in the form
u,(x,t) = tpi(t)
W(x) + Lpi(t)
, v(x,t) =1p2(t)
W(x)+V2(t)
(1.4)
= x) + (B, x) + C, (1.5)
where ipi(t), <Pi(t), i = 1,2 are unknown functions of time; a nonzero numerical symmetric matrix A of dimension n x n, a constant vector B £ R" and a constant C € R will be defined later. Here and in what follows (•, •) is the scalar product in Rra. Formulas similar to (1.4)—(1.5) were successfully used by the authors in several previous papers [10-12] to construct exact solutions of the equation of nonlinear heat conduction and of nonlinear
parabolic systems with exponential nonlinearities [5; 6]. Note that the ideology to construct solutions in the form (1.4) corresponds to the method of generalized separation of variables [2;3;9; 15; 16].
2. Reduction to a system of ODE
After the substitution of the functions (1.4) into the system of equations (1.1) and simple calculations, we obtain the equations
VWx) + (VW = ß2mi\VW(-x.)\2 + a2ml
W(x)+ip2
W(x) + tpi
+ №, (2-1)
+g(t). (2.2)
Here, tpi = ipi(t), (fi = <pi(t), ip'i = immediately find
dipj dt
dipi dt
i = 1,2. From (1.5) we
|VIF(x)|2 = (A2x,x) +2(AB,x) + |B|
In view of these relations and (1.5) the equalities (2.1), (2.2) can be rewritten as
^(Ax,X) + (B,X) + C +&№)' =
(V4 - 04(^2)
ß1 [(A2X, x) + 2(AB, x) + |B|2] + ai(t)ip2(P2 + №, (2.3)
-(Ax,x) + (B,x)+C
+ (tp2<f2)' =
I32m22 [(¿2x, x) + 2(AB, x) + |B|2] + a2(i)Vw 1 + g(f). (2.4)
It is easy to verify that if the symmetric matrix A, the vector B and the constant C satisfy the following system of algebraic equations (SAE):
A = 2(7 A2, B = 2(TAB, C = <T|B|
(2.5)
where a / 0 is the separation constant, then the equalities (2.3), (2.4) are reduced to the following system of ordinary differential equations (ODE):
ßi{t)
Â(t)
V4 " - ^^ n = 0, 1P2 - ^^ r2= 0
(2.6)
a a
(ipi<Pi)' - ai(t)ip2<P2 ~ f(t) = 0, (ip2<P2)' ~ a2(t)ipi(pi -g(t) = 0. (2.7) Hence, the following statement is valid.
Theorem 1. The nonlinear system of first order partial differential equations (1.1) has exact solutions (1.4), where the function И^(х) can be an arbitrary polynomial of the form (1.5) with the coefficients satisfying the SAE (2.5), and the functions ipi(t), <fi(t), i = 1,2 are solutions of ODE systems (2.6), (2.7).
Remark 1. Theorem 1 allows us to decompose the construction of the solution's components depending on spatial variables and time. Such a decomposition essentially simplifies the problem in that it replaces the original nonlinear system of first order partial differential equations by systems of algebraic and ordinary differential equations. This may prove useful in the development of numerical methods and algorithms for constructing approximate solutions of the corresponding boundary value problems.
The algebro-differential systems (2.5)-(2.7) consist of a block of algebraic equations (2.5) and a block of ODE (2.6), (2.7). We will investigate each block separately.
3. Solvability of algebraic equations
First of all, we consider the solvability of the matrix equation of the system (2.5). We note that it always has the trivial solution A = 0. Hence, in the sequel we consider only nontrivial solutions of this equation. It is easy to verify that a solution of the matrix equation of system (2.5) is of
the form A = — P, where P is an arbitrary idempotent matrix, i.e. a
matrix satisfying the equality P2 = P. It is known [4] that any idempotent matrix P can be written as P = MEmM~l, where M is an arbitrary non-degenerate matrix of order n, Em is a diagonal matrix with the diagonal having m € {1,2,... ,n} units and n — rri zeros arranged in an arbitrary order; Em is also an idempotent matrix: E^ = Em. Since we are interested only in symmetric matrices A, we need to choose idempotent matrices P to be symmetric as well, i.e. P = SEmST, where 5 is an arbitrary orthogonal matrix. Therefore, the matrix
A = ^SEmST (3.1)
2(7
is a solution of the matrix equation (2.5). In this case, we have
171
tr A = —, m<n, n € N, n > 2. (3.2)
2(7
The vector equation (2.5) is a system of n linear homogeneous algebraic equations with respect to the components b\,..., bn of the unknown vector B. For any fixed matrix (3.1) with the rank rank A = m < n there always
exists a nontrivial solution to the linear homogeneous system. Moreover, the components b\,..., bm of the vector B can be chosen arbitrarily from an m -dimensional linear manifold. In the case when rank A = m = n, i.e. if Em = E, the linear homogeneous system of equations has a solution, which is an arbitrary vector B € Rra. Once the solutions of the matrix and vector equations are successively found, the constant C is uniquely determined from the scalar equation of the system (2.5).
Example 1. Let n = 3 and square 3x3 matrices Em have ranks 3, 2, and 1, respectively. An arbitrary orthogonal matrix S of dimension 3x3 is represented by the formula
5 =
an — sin(si) cos(s2) ai3
a2i cos(si) cos(s2) a23
cos(s2) sin(s3) — sin(s2) cos(s2) cos(s3)
Here, for simplicity we introduce the following notation
an = cos(si) cos(s3) — sin(si) sin(s2) sin(ss),
a2i = cos(si) sin(s2) sin(s3) + sin(si) cos(ss), ai3 = — cos(si) sin(s3) — sin(si) sin(s2) cos(ss), a23 = cos(si) sin(s2) cos(s3) — sin(si) sin(ss). To construct specific examples, we give the parameters si, s2, S3 certain numerical values. Let, for instance, si = —s2 = arcsin s3 =
then the orthogonal matrix S will have the form
5 =
-1/3 2/3 -2/3 2/3 2/3 1/3 2/3 -1/3 -2/3
The matrix Em of rank 3 coincides with the identity matrix and, in this case, we obtain the form - (Ax, x) = — ( x2 + y2 + z2 ]. For the matrix A
2 4(7 V J
of this quadratic form a solution of the vector equation of the system (2.5) is an arbitrary vector B = (&i,&2,&3). In addition, formula (1.5) will have the form
Wo(x, y,z) = (x2 + y2 + z2) + bix + b2y + b3z + a (b\ + b\ + bf) (3.3)
and we obtain solutions radially symmetric with respect to the spatial variables x, y, z.
Now, take the matrices Em of rank 2:
Elm =
For these matrices, from (3.1) we have 1
" 1 0 о" "1 0 о" "0 0 о"
0 1 0 , Е2т — ООО , Езт — 0 1 0
0 0 0 0 0 1 0 0 1
A, =
18(7
5 2-4 2 8 2 -4 2 5
A, =
18(7
18(7
8 2 2 2 5-4 2-4 5
5-4 2 -4 5 2 2 2 8
From the vector equation (2.5) we obtain the corresponding vectors Bi, B2, B3:
B1 = (l1,2l1+2k,h), B2 = (h,l2,2h + 2l2), B3 = (2l2 + 2h,l2,h),
where /1, l2, /3 are arbitrary constants. Hence, the constants C1, C2, C3 are given by the formulas
C\ = (7 Ij + 8/1/3 + 5/2^, C2 = (7 (5Ij + 8/1/2 + 5/|), C3 = (7 (5/22 + 8/2/3 + 5/32).
Consequently, we obtain Wi(x,y,z) =
1
180(7
[(5ж + 2y - 4z)2 + 9(2y + z)2] + (3.4)
+ l\x + (2/i + 2/3)2/ + hz + a (5ij + 8/1/3 + 5/f), W2(x, y,z) = y^ [(5a; - Ay + 2z)2 + 9(y + 2z)2] + (3.5) + hx + l2y + (2/1 + 2/2) 2 + (7 (5/f + 8/1/2 + 5/2) , W3(x, = ^ [(4x + y + z)2 + 9(y - z)2] + (3.6)
+ (2/2 + 2/3) x + l2y + l3z + (7 (5/| + 8/2/3 + 5if).
Therefore, in this case, we have solutions anisotropic with respect to the spatial variables x, y, z.
Finally, consider the matrices Em of rank 1:
' 1 0 о" "0 0 о" "0 0 о"
E\m — 0 0 0 , Е2т — 0 1 0 , Езт — 0 0 0
0 0 0 0 0 0 0 0 1
For these matrices, from (3.1) we have
Ai =
1
18(7
1 -2 -2 -2 4 4 -2 4 4
A, =
Ao =
1
№t
4 4-2 4 4-2 -2 -2 1
18(7
4-2 4 -2 1 -2 4-2 4
From the vector equation (2.5) we obtain the corresponding vectors Bi, B2, B3:
B1 = (1,-21,-21), B2= [1,1,-^1
B 3=[l,~\l,l),
where I is an arbitrary constant. Hence, the constants C\, C2, C3 are given
9
9
by the formulas C\ = 9al , C2 = - al , C3 = - al . Finally, we get
W\(x, y, z) =-u)\ + lu) 1 + 9al2, where u)\ = x — 2y — 2z,
36(7
1 19
W2(x, y, z) = —- ujI + -(jj2 + - al2, where w2 = 2x + 2y - z, (3.7) 36(7 2 4
W3(x,y,z) =
1
I 9
uj2 + - W3 + - al2, where W3 = 2x — y + 2z.
36a 0 2 4
In this case, we will actually obtain solutions j ¡pseudo-multidimensiona^ with respect to the variables uji, uj2, ujs, i.e. solutions containing a linear combination of spatial variables.
For n = 2, square 2x2 matrices Em of unit rank also provide ¡¡pseudo-multidimensional^^ solutions. And the identity matrix Em of rank 2 for
any 2x2 orthogonal matrix S gives W(x, y) = — (x2 + y2) + b\x + b2y +
4(7
a(b\ + b2), where b\, b2 are arbitrary constants. In this case, we have solutions radially symmetric with respect to the spatial variables x and y. Therefore, in the two-dimensional coordinate space, the nonlinear parabolic system (1.1) does not have multidimensional exact solutions anisotropic with respect to the spatial variables of the form (1.4) with the function (1.5).
4. Solvability of the ODE system (2.6), (2.7)
In this section, we construct exact solutions of the ODE system (2.6), (2.7) under the assumption that the functions «¿(i), fii(t) are identically
constant: Oj(t) = щ € К, щ ф 0, fa{t) = fa € К, fa ф 0, % = 1, 2. Such an assumption is useful when searching solutions of the system of equations (2.6) in the case when 0i(t), 02 (i) are constant and when constructing periodic in time solutions of the system (1.1).
4.1. The case when the functions 0i(t), 02(i) are constant
We start with the integration of the nonlinear ODE system (2.6) with respect to the unknown functions 0i(t), 02(i). But, first, we note that nontrivial constant solutions of the ODE system (2.6) are equally interesting for our study. They are determined from the following system of nonlinear algebraic equations:
Pi fa
— ф\ + Oi02 = 0, — + o20i = 0. a a
Whence we find a nontrivial solution
( aja2\1/3 a fa ( ofo2\2/3 . .
For these constants ipw, Ф20 the ODE system (2.7) will have the form
■010^1 - «1020^2 = f(t), 020^2 - «2010^1 = g(t), (4.2)
Hence, taking into account the above results and using Theorem 1 we see that the following statement holds.
State 1. The nonlinear system of first order partial differential equations (1.1) has an exact multidimensional solution
U(X, t) = 010 [*F(X) + Lp\(t)] , U(X, t) = 020 [iF(x) + <p2(t)] ,
where the constants 0ю, 020 are given by (4.1), and the functions <fi(t), (p2(t) are determined from the linear ODE system (4.2).
Example 2. Let 0:10:2 > 0, then the nonlinear system of first order partial differential equations of the form
ut = fa\Vu\2 + aiv + sin(t), vt = fa I Vf|2 + a2u + cos (t),
in the three-dimensional coordinate space, has a parametric family of particular exact solutions anisotropic in the spatial variables
uk(x,y,z,t) = 010 [Wk(x,y,z) +(fi(t)} , к = 1,2,3,
vk(x, y, z, t) = 020 [Wk(x, y, z) + <fi2(t)] , k = 1,2,3,
where the constants -010, ^20 are given by (4.1), Wk(x,y, z), k = 1,2,3 are defined by the formulas (3.4)—(3.6), and the functions <pi(t), tp2(t) have the form
Oil + 1
<P!(t) = Cxe-^^1 + Cze^^1 - -— 1 , , cos(t),
{aia2 + l)^io
^(t) = p- (C2e^t - + J-^- _L Sin(i).
a 1 ^20 ^ ' OL\a2 +1 V>20
Here, C1, C2 are arbitrary constants. Example 3. (Periodic in time solutions)
Let a\ = —p2, a2 = q2, p / 0, q / 0 be arbitrary parameters, f(t) = g(t) = 0. Then, the nonlinear system (1.1), in the three-dimensional coordinate space, has a parametric family of particular periodic in time exact solutions anisotropic in the spatial variables
uk(x, y, z, t) = ipw [Wk(x, y, z) + tpi{t)\ , k = 1,2,3,
vk(x, y, z, t) = ip20 [Wk(x, y, z) + tp2{t)\ , A; = 1,2,3,
where the constants ^10, ^20 are given by (4.1), Wk(x,y, z), k = 1,2,3 are defined by the formulas (3.4)—(3.6), and the functions <pi(t), tp2(t) have the form
(pi(t) = Ci s'm(pq t)+C2 cos(pq t), ip2(t) = - (C2 s'm(pqt)-Ci cos(pqt)).
P W20
Here, C1, C2 are arbitrary constants.
4.2. The case of non-constant functions ipi(t), ip2(t)
The ODE system (2.6) is easily reduced to one nonlinear ODE of the second order with respect to the function ipi(t) of the following form:
< - — V42 + 2Jl f-^-V? -V>iVi - -«1^1 = 0.
a i<r a \ai(T / ai<Td
By the substitution ipi(t)' = y(ipi) this ODE is reduced to the following Abel equation of the second kind
dy fi2 2Wl(fi2 2 \ I32I32 4
y—---y +- -ipl-ipi)y--3 ip1 - aia2ipi = 0.
dip 1 a\a a \aicr J a\aa
Let the parameters of the original systems are related by the equality (3fai + /?|o;2 = 0. Then, the Abel equation has a particular exact solution y(ipi) =
(h , 2 fii&i i Pitti<J ^ ^ eag g^ain a particular exact
a fa /?2
solution of the system (2.6) having the form
№) = ^ (73tan(T) - l) , 02*(i) = ^f (V3 tan(T) + l) . (4.3) j3 Qi v^
Here, we denote T = ——— (t — to), where to is an arbitrary constant.
2p2
Hence, given the results above invoking Theorem 1 we have the following statement.
State 2. The nonlinear system of first order partial differential equations (1.1) with the parameters related by the equality fifai + /?2«2 = 0 has an exact multidimensional solution
u(x,t) = 01* (t) [W(x) + <Pl (*)] , u(x,i) = 02* (t) [W(x) + <p2{t)\ ,
where the functions ipl(t), ijj2{t) are given by (4.3), and the functions <fi(t), <p2(t) are determined from the system of linear non-autonomous and inhomogeneous ODE (2.7).
We note that in Statements 1, 2, in the case n = 3 (x € R3), as W(x, y, z) we can take any of the functions given in Example 1. Thus, for the functions defined by (3.4)-(3.6) we obtain exact solutions anisotropic with respect to the spatial variables, and for the function (3.3) we can write down an exact solution radially symmetric with respect to the spatial variables x, y, z
Example 4. The nonlinear system of first order partial differential equations of the form
ut = fh\Vu\2 + v + sh(t), vt = fo\Vv\2 -u- ch (t),
with the parameters such that fi2 — /3| = 0 has a parametric family of particular exact solutions anisotropic with respect to the spatial variables
uk(x, y, z, t) = iii (V3tan(T) - l) [Wk(x, y, z) + <pi(t)] , k = 1,2,3, vk(x, y, z, t) = ¡i2 (V3 tan(T) + l) [Wk(x, y, z) + ip2(t)] , k = 1,2,3,
where /xi = /x2 = Wk(x,y,z), k = 1,2,3 are defined by (3.4)-
Zp2 ¿p2
(3.6), and the functions <£>i(t), <f2(t) have the form
C\ sin t + C2 cost ¡jl\ (Ci cost — C2 sin t) — sht
Vlit)= \/3tg(T) -1 ' V2it)= ,2 (73tg(T) +1) • Here, sht, cht are the hyperbolic sine and cosine, respectively, to, C1, C2
are arbitrary constants, T = ——— (t — to).
2p2
5. Exact solutions of the Hamilton-Jacobi type equation
In addition, we construct exact multidimensional solutions of the Hamilton-Jacobi type equation (1.2). Namely, for the equation (1.2) we have
State 3. The Hamilton-Jacobi type equation (1.2) has an exact multidimensional solution
u(x, t) = 1p(t)
^(Ax, x) + (B, x) + C + ip(t)
(5.1)
where the functions ip(t), <p(t) satisfy the ODE system
tf - a(t)ip - ® = 0, №<p)' = a(t)ipip + f(t), (5.2) a
and the symmetric matrix A, the constant vector BetR" and the constant C € IR satisfy SAE (2.5) with an arbitrary constant a / 0.
The validity of this statement can be verified by the direct substitution of the function (5.1) to the equation (1.2). After elementary calculations in view of the equalities (2.5) we arrive at the ODE system (5.2). Note that the function (5.1) is also a solution of Eq. (1.3). Some of the exact solutions of Eq. (1.3) given in [1] are obtained for the function (5.1) as a particular case.
Example 5. The Hamilton-Jacobi type equation of the form
ut = f3t\Vu\2 + ju + tx,
has a parametric family of exact solutions anisotropic with respect to the spatial variables:
uk(x, y, z, t) = ip(t) [Wk(x, y, z) + Lp{t)} , k = 1,2,3,
where Wk(x,y, z), k = 1,2,3 are defined by (3.4)—(3.6), and the functions ip(t), <p(t) have the form
/ u\ a + 2
#0 =
C\(a + 2)t~a — fit2 '
^A+l—a
/ fX+1-a \
m = (Ci(a + 2) - (c2 - (g + 2)(a_A.1)) ■
Here, a / —2, /3/0, A/a — 1 are arbitrary parameters of the equation, Ci, C2 are the constants of integration.
6. Conclusion
In the present paper, we obtain formulas for new exact multidimensional solutions of the nonlinear system of first order partial differential equations (1.1) using the construction (1.5), which has never been used before by other authors. From the point of view of finding exact multidimensional solutions, a useful feature of the construction (1.5) is that along with the
linear summands it contains the quadratic form of n variables -(Ax, x).
From our results it follows that the spatial structure of the solutions is determined by the rank of the matrix A. If rank A = 1, then we have ¡¡pseudo-multidimensional^^ exact solutions, i.e. solutions with a linear combination of the spatial variables. If 1 < rank A < n, then we obtain exact solutions anisotropic with respect to the spatial variables. Finally, if rank A = n, then we have exact solutions radially symmetric with respect to the spatial variables. It is shown that in the two-dimensional coordinate space the nonlinear system of first order partial differential equations (1.1) does not have multidimensional exact solutions anisotropic with respect to the spatial variables of the form (1.4) with the function TF(x) of the form (1.5). Exact multidimensional solutions of the Hamilton-Jacobi type equation (1.2) are constructed. The explicit expressions of the exact multidimensional solutions obtained in the article, which are expressed in terms of elementary functions, are important both from the theoretical and applied points of view, as they can be used for testing, adjusting and adapting numerical methods and algorithms for constructing approximate solutions of boundary value problems for nonlinear systems of first order partial differential equations of a large dimension.
References
1. Fushhich V.I., Shtelen V.M., Serov N.I. Simmetrijnyj analiz i tochnye reshenij nelinejnyx uravnenij matematicheskoj fiziki [Symmetry analysis and exact solutions of nonlinear equations of mathematical physics]. Kiev, Naukova dumka, 1989, 336 p. (in Russian)
2. Galaktionov V.A. , Posashkov S.A. New exact solutions of parabolic equations with quadratic nonlinearities. Comput. Math. Math. Phys., 1989, vol. 29, no. 2, pp. 112-119.
3. Galactionov V.A., Svirshchevskii S.R. Subspaces of nonlinear partial differential equations in mechanics and, physics. Chapman & Hall/CRC, 2007, 493 p.
4. Gantmaxer F.R. Teoriya matric [Matrix Theory]. Moscow, Nauka Publ., 1988, 552 p. (in Russian)
5. Kosov A.A., Semenov E.I. Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms. Siberian Math. J., 2017, vol. 58, no. 4, pp. 619-632. https://doi.org/10.1134/S0037446617040085
6. Kosov A.A., Semenov E.I. On Exact Multidimensional Solutions of a Nonlinear System of Reaction-Diffusion Equations/Differential Equations, 2018, vol.54, no 1, pp. 106-120. https://doi.org/10.1134/S0012266118010093
7. Kurzhanski A.B. Comparison principle for equations of the Hamilton-Jacobi type in control theory. Proceedings of the Steklov Institute of Mathematics, 2006, vol. 253, supplement 1, pp. 185-195. https://doi.org/10.1134/S0081543806050130
8. Markov Y., Rudykh G., Sidorov N., Sinitsyn A., Tolstonogov D. Steady state solutions of the Vlasov-Maxwell system and their stability. Acta Appl. Math., 1992, vol. 28, no. 3. pp. 253-293.
9. Polyanin A.D., Zajcev V.F., Zhurov A.I. Metody resheniya nelinejnyx uravnenij matematicheskoj fiziki i mexaniki [Methods for solving nonlinear equations of mathematical physics and mechanics]. Moscow, FIZMATLIT, 2005, 256 p. (in Russian)
10. Rudykh G.A., Semenov E.I. The construction of exact solutions of the multidimensional quasilinear heat-conduction equation. Comput. Math. Math. Phys, 1993, vol. 33, no. 8, pp. 1087-1097.
11. Rudykh G.A., Semenov E.I. Exact nonnegative solutions of the multidimensional nonlinear diffusion equation. Siberian Math. J., 1998, vol. 39, no. 5, pp. 977-985. https://doi.org/10.1007/BF02672920
12. Rudykh G.A., Semenov E.I. Non-self-similar solutions of multidimensional nonlinear diffusion equations. Math. Notes, 2000, vol. 67, no. 2, pp. 200-206. https://doi.org/10.1007/BF02686247
13. Sidorov N.A., Sinitsyn A.V. Stacionarnaya sistema Vlasova - Maksvella v ogranichennyx oblastyax. [The stationary Vlasov - Maxwell system in bounded domains]. Nelinejnyj analiz i nelinejnye differencialnye uravneniya [Nonlinear analysis and nonlinear differential equations]. Moscow, Fizmatlit Publ., 2003, pp. 50-88. (in Russian)
14. Skubachevskii A.L. Vlasov - Poisson equations for a two-component plasma in homogeneous magnetic field. Russian Math. Surveys, 2014, vol. 69, no. 2, pp. 291335. http://dx.doi.org/10.1070/RM2014v069n02ABEH004889
15. Titov S.S. Metod konechnomernyh kolec dlya resheniya nelinejnyh uravnenij matematicheskoj fiziki [The method of finite-dimensional rings for solving nonlinear equations of mathematical physics] Aerodinamika, Saratov, Saratovskij Univ. Publ., 1988, pp. 104-110. (in Russian)
16. Zajcev V.F., Polyanin A.D. Spravochnik po differencialnym uravneniyam s chast-nymi proizvodnymi pervogo poryadka [Handbook of differential equations with partial derivatives of the first order]. Moscow, Fizmatlit Publ., 2003, 416 p. (in Russian)
Alexander Kosov, Leading Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Post Box 292, 134, Lermontov st., Irkutsk, 664033, Russian Federation, tel.: (3952) 427100 (e-mail: kosov_idstu@mail.ru)
Edward Semenov, Senior Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Post Box 292, 134, Lermontov st., Irkutsk, 664033, Russian Federation, tel.: (3952) 453099 (e-mail: edwseiz@gmail.com)
Vladimir Tirskikh, Associate Professor, Irkutsk State Transport University, 15, Chernyshevsky st., Irkutsk, 664074, Russian Federation, tel.: (3952) 638311 (e-mail: tirskikh_vv@irgups.ru)
Received 17.04.19
О точных многомерных решениях одной нелинейной системы уравнений с частными производными первого порядка.
A. А. Косов
Институт динамики систем и теории управления им. В. М. Ма-тросова СО РАН, Иркутск, Российская Федерация
Э. И. Семенов
Институт динамики систем и теории управления им. В. М. Ма-тросова СО РАН, Иркутск, Российская Федерация
B. В. Тирских
Иркутский государственный университет путей сообщения, Иркутск, Российская Федерация
Аннотация. В статье изучается система двух нелинейных уравнений в частных производных первого порядка. Правые части системы уравнений содержат квадраты градиентов искомых функций. Такого рода уравнения, близкие к уравнению Гамильтона - Якоби, встречаются в задачах механики и теории управления. В статье предлагается искать решение в виде анзада, содержащего квадратичную зависимость от пространственных переменных и произвольные функции от времени. Использование предложенного анзада позволяет декомпозировать процесс отыскания компонент решения зависящих от пространственных переменных и от времени. Для отыскания зависимости от пространственных переменных необходимо решать алгебраическую систему матричных, векторных и скалярного уравнения. Найдено общее решение этой системы уравнений в параметрическом виде. Для отыскания компонент решения исходной системы, зависящих от времени, возникает система нелинейных дифференциальных уравнений. Установлено существование точных решений определенного вида у исходной системы. Приводится ряд примеров построенных точных решений, в том числе периодические по времени и анизотропные по пространственным переменным. Проведен анализ пространственной структуры решений, установлено, что она зависит от ранга матрицы квадратичной формы, входящей в решение.
Ключевые слова: нелинейная система, уравнения типа Гамильтона - Якоби, точные решения.
Список литературы
1. Фущич В. И., Штелень В. М., Серов Н. И. Симметрийный анализ и точные решений нелинейных уравнений математической физики. Киев : Наукова думка, 1989. 336 с.
2. Галактионов В. А., Посашков С. С. О новых точных решениях параболических уравнений с квадратичными нелинейностями // Журн. вычисл. математики и мат. физики. 1989. Т. 29, № 4. С. 497-506.
3. Galactionov V. A., Svirshchevskii S. R. Subspaces of nonlinear partial differential équations in mechanics and physics. Chapman & Hall/CRC, 2007. 493 p.
4. Гантмахер Ф. P. Теория матриц. M. : Наука, 1988. 552 с.
5. Косов А. А., Семенов Э. И. О точных многомерных решениях системы уравнений реакции-диффузии со степенными нелинейностями // Сиб. мат. журн. 2017. Т. 58, №4. С. 796-812. https://doi.org/10.17377/smzh.2017.58.408
6. Косов А. А., Семенов Э. И. О точных многомерных решениях одной нелинейной системы уравнений реакции-диффузии // Дифференц. уравнения. 2018. Т. 54, № 1. С. 108-122. https://doi.org/10.1134/S0374064118010090
7. Куржанский А. Б. Принцип сравнения для уравнений типа Гамильтона — Якоби в теории управления // Тр. ИММ УрО РАН. 2006. Т. 12, № 1. С. 173-183.
8. Markov Y., Rudykh G., Sidorov N., Sinitsyn A., Tolstonogov D. Steady state solutions of the Vlasov-Maxwell system and their stability // Acta Appl. Math. 1992. Vol. 28, N 3. P. 253-293.
9. Полянин A. Д., Зайцев В. Ф., Журов А. И. Методы решения нелинейных уравнений математической физики и механики. М. : Физматлит, 2005. 256 с.
10. Рудых Г. А., Семенов Э. И. Построение точных решений многомерного уравнения нелинейной теплопроводности // Журн. вычисл. математики и мат. физики. 1993. Т. 33, № 8. С. 1228-1239.
11. Рудых Г. А., Семенов Э. И. Точные неотрицательные решения многомерного уравнения нелинейной диффузии // Сиб. мат. журн. 1998. Т. 39, № 5. С. 1131— 1140. https://doi.org/10.1007/BF02672920
12. Рудых Г. А., Семенов Э. И. Неавтомодельные решения многомерного уравнения нелинейной диффузии // Мат. заметки. 2000. Т. 67. № 2. С. 250-256. https://doi.org/10.4213/mzm833
13. Сидоров Н. А., Синицын А. В. Стационарная система Власова-Максвелла в ограниченных областях // Нелинейный анализ и нелинейные дифференциальные уравнения. М. : Физматлит, 2003. С. 50-88.
14. Скубачевский A. JL Уравнения Власова - Пуассона для двухкомпонентной плазмы в однородном магнитном поле // Успехи мат. наук. 2014. Т. 69, № 2. С. 107-148. https://doi.org/10.4213/rm9579
15. Титов С. С. Метод конечномерных колец для решения нелинейных уравнений математической физики // Аэродинамика. Саратов : Саратов, ун-т, 1988. С. 104-110.
16. Зайцев В. Ф., Полянин А. Д. Справочник по дифференциальным уравнениям с частными производными первого порядка. М. : Физматлит, 2003. 416 с.
Александр Аркадьевич Косов, кандидат физико-математических наук, ведущий научный сотрудник, Институт динамики систем и теории управления им. В. М. Матросова СО РАН, Российская Федерация, 664033, г. Иркутск, ул. Лермонтова, 134, тел.: (3952) 427100 (e-mail: kosov_idstu@mail.ru)
Эдуард Иванович Семенов, кандидат физико-математических наук, старший научный сотрудник, Институт динамики систем и теории управления имени В.М. Матросова СО РАН, Российская Федерация, 664033, г. Иркутск, ул. Лермонтова 134, тел.: (3952) 453099 (e-mail: edwseiz@gmail.com)
Владимир Викторович Тирских, кандидат физико-математических наук, доцент, Иркутский государственный университет путей сообщения, 664074, г. Иркутск, ул. Чернышевского, 15, тел.: (3952)638311 (e-mail: tirskikh_vv@irgups.ru)
Поступила в редакцию 17.04-19
