The non-autonomous dynamical systems and exact solutions with superposition principle for evolutionary PDEs Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том 4. № 4 (2012). С. 186-195.

УДК 517.9

THE NON-AUTONOMOUS DYNAMICAL SYSTEMS AND EXACT SOLUTIONS WITH SUPERPOSITION PRINCIPLE FOR EVOLUTIONARY PDEs

V.A. DORODNITSYN

Abstract. In the present article we introduce a new application of S. Lie’s non-autonomous dynamical systems with the generalized separation of variables in right hand-sides. We consider non-autonomous dynamical equations as some sort of external action on a given evolution equation, which transforms a subset of solutions into itself. The goal of our approach is to find a subset of solutions of an evolution equation with a superposition principle. This leads to an integration of ordinary differential equations in a process of constructing exact solutions of PDEs. In this paper we consider the application of the most simple one-dimensional case of the Lie theorem.

Key words: evolutionary equations, exact solutions, superposition of solutions.

Introduction

The concept of linear superposition of solutions in classical theory of linear ordinary differential equations

Xt(t) = фпфх1 + ... + фт(^ХП, i = 1, 2, ...,n,

was generalized by Sophus Lie [1] for nonlinear dynamical systems with the generalized separation of variables in right hand-sides. Namely, S.Lie proved the following theorem. Theorem. The equations

xt(t) = fг{t,x), i = 1,

possess a fundamental set of solutions, i.e. its general solution can be represented by finite number m of particular solutions x^, ...,x™; ...,x]n, ...,xr^n and n number of arbitrary constants C1, ...,Cn, if and only if they have the following form

Зтг

— (t) = + ... + фг (tfCr {x), i = 1, 2,...)П) (1)

where the coefficients £га(х) satisfy the condition that the operators

d

Xa = Ca{x) —, a =1, 2,...,r, (2)

span a Lie Algebra Lr of a finite dimension r. The number m of necessary particular solutions is estimated by inequality mn > r. The superposition formulae for a general solution

д Ч г

хг = S%(x1,...,xn; x\,...,xn; ...,xxm, ...,xnm\ G\,...,Cn), i = 1, 2,...,n, det || ——1| = 0.

Obk

V.A. Dorodnitsyn, The non-autonomous dynamical systems and exact solutions with superposition principle for evolutionary PDEs.

© V.A. Dorodnitsyn 2012.

The research was sponsored in part by the Russian Fund for Basic Research under the research project No. 12-01-00940-a.

Поступила 27 октября 2012 г.

are defined implicitly by n equations

dJi

Ji(xl,...,xn; x\,...,x1; ...,xlm, ...,xr!n) = Ch i = 1, 2, ...,n, det ||^|| = 0,

where Ji are functionally independent with respect to x% invariants of the operators (2)

prolonged to the (n + mn)-dimensional space

d ■ d ■ d Xa = CM -Q— + Cai^l) + ... + Cai^m) ~q— , « = 1, 2, ...,T.

The non-autonomous dynamical system (1) will be referred as the Lie non-autonomous

dynamical system (NADS) or the Lie system. Notice, that the statement of the Lie theorem has various aspects. The first one states that the superposition (which is nonlinear in general) of finite number particular solutions is again a solution. The second one is that its general solution can be represented by finite number m of particular solutions. Thirdly, the Theorem based on an invariant object which is Lie Algebra of operators Xa.

Notice, that if all functions fai = const, i = 1, 2,...,n, then system (1) simply represents a one-parameter Lie group of point transformations, which generated by linear combinations (with constant coefficients fa) of operators (2). For variable coefficients fa(t) the system (1) is sufficient different from a one-parameter Lie group of point transformations.

In 1980-th there was the renewed interest to this Lie’s theorem, and several important applications were found [2,3,4,5]. The discussion of Lie’s theorem and several examples of applications one can find in [6].

In this paper we consider the new application the above theorem. We do not investigate system of type (1) itself, but consider non-autonomous equations as some sort of external action on some given evolution equation. The goal of our approach is to find a subset of solutions of evolution equation which possess the superposition principle. Solutions of a non-autonomous equation will be considered as some generalization of symmetry transformations, which act on an evolution equation and transform a subset of solutions into itself. This leads to an integration of ordinary differential equations in a process of developing exact solutions of PDEs. We supply the theory with examples.

The article is organized as following. In Section 1 we formulate the most simple onedimensional case of the Lie theorem. Section 2 devoted to the technique for constructing special solutions with linear superposition principle, what demonstrated on examples in Section 3. In Section 4 we generalize the approach for evolutionary PDEs in two space dimensions. Sections 5 and 6 devoted to solutions with the Bernoulli and the Riccati types superpositions. In final Section 7 we develop the necessary and sufficient conditions for evolutionary PDEs to possess subset of solutions with linear superpositions principle.

1. One-dimensional case of the Lie’s theorem

For n = 1 the most general Lie’s non-autonomous dynamical equation is the Riccati equation

ut(t) = fa(t) + fa(t)u + fai(t)u2, (3)

where fai,i = 1, 2, 3 are some smooth functions of t.

The equation (3) possesses the fundamental set of solutions as far it is associated with the Lie Algebra L3

d d d

X\ = —, X2 = u—, X2 = u2 —, ou ou ou

of the projective group. For the equation (3) n = 1,r = 3,m > 3. In fact three particular

solutions casts the minimum number of solutions to develop a general solution of Riccati

equation.

The subalgebras of (3) are casted by 1-dimensional subalgebras spanned by each operators individually and by two-dimensional subalgebras X2) and (X2; X3). The equation (3)

possesses the well-known non-linear superposition principle for their solutions

(uA - U3)(U2 - Ui) _ c (uA - щ)(щ - Ui)

where С is an arbitrary constant, and its general solution can be expressed by means of three particular solutions

_ U3(U2 - Ui) - Сщ(щ - Ui)

u2 - u1 - С(u3 - u1)

For the 1-dimensional subalgebras spanned by each operators Xi, X2; X3 individually there exists point transformations, which change corresponding dynamical equations into classical Lie group equations. Thus, nontrivial cases start from two-dimensional subalgebras (Xi, X2) and (X2; X3).

2. The subset of solutions with linear superposition

д d

We firstly consider the subalgebra L2 spanned by the operators X1 _ —-, X2 _ u—. The

Ou Ou

corresponding non-autonomous evolution is a linear equation

ut(t)_ фі(ї) + Ф2(t)u. (4)

For equation (3) n _ l,r _ 2,m > 2. In fact two particular solutions casts the minimum number of solutions to develop a general solution of the equation (4). Thus, (4) has a fundamental set of special solutions u1 and u2 with superposition

и - u1

C,

U2 — U\

which yield the general solution as

и = (1 — С )u\ + Cu2, С = const. (5)

In this paper we consider non-autonomous equations as some sort of external action on a given evolution equation. Following the idea of classical Lie group analysis of differential equations [7,8] we involve the prolongation of non-autonomous dynamical system for spatial derivatives.

Thus, we involve x as one more independent variable and let и be dependent on two variables: и = u(t,x). We rewrite (4) in the form

ut(t, x) = ф(ї) + ґф'(t)u(t, x),

(we write vp'(t) for convenience) and prolong it for evolution of partial derivative ux,uxx,...:

Ut(t,x) = ф(ї) + ґф' (t)u(t,x), (6)

Uxt(t,x) = i>'(t)Ux(t,x), Uxxt(t,x) = i>'(t)Uxx(t,x),. . . .

The above evolutionary system can be associated with the Lie algebra L2 spanned by the following operators:

5 5 д d

X\ = —, X2 = и——+ иx— + Uxx-гг- ,....

UU UU UUx OUxx

Notice, that for the system (6) we still have n = 1, r = 2, as far as all differential sequences of the first evolution equations do not produce new dependent variables. One can write down a solution of the linear equations (6) in the following form

u(t,x) = ф(і) + еф(і)У (x),

where V(x) describes dependence of u on space variable x, and fa(t) is a special solution of inhomogeneous equation

fa = fa(t)+ (t)fa.

Notice, that the superposition formula (5) yields the superposition for V(x) as well:

V = (1 - C)Vi + CV2,

where C = const.

Thus, the set of solutions (Vi,V2) span a linear space.

Now we consider an evolutionary equation

Ut — F(U, Ux, Uxx) , (7)

which, in general, is nonlinear. We consider the compatibility condition for evolution (6) and evolutionary eq. (7), that gives ODEs for unknown functions fa(t),fa(t).

Now we consider a row of examples of evolutionary equations, when such a compatibility exists. Most of the following examples of equations were taken from [9,10,11,12,13] and yield subspaces of solutions with various dimensions. In all cases we have linear superposition and two functions fa(t),fa(t) to describe the corresponding subset independently of space dimension.

3. Examples of evolutionary equations with a linear subspace of solutions

Example 1. We consider the nonlinear equation

22

Ut — Uxx + ^x — ^ ,

and looking for the special solution in the form

u(t,x) = fat) + e^{t)V (x), Ux(t,x) = emVx(x), Uxx(t,x) = e^(t)Vxx(x), (8)

which yields the following equation

fa + ^e* V = e* V'' + e2 * V'2 — fa — 2faefa V — e2 ^ V2.

Splitting the last equation gives

fa = —fa2 + e2^ ((V')2 — V2), fa'V = V'' — 2faV.

Then we have the following overdetermined system

(V')2 — V2 = k = const, V" = mV, m = const.

From the last relations in the case k = 0 we have

m =1, V (x) = Aex + Be-x, k = —4AB.

So, we can write down the family of solution as

u(t, x) = fat) + em (Aex + Be-x),

where functions fa(t),fa(t) satisfy the following dynamical system:

fa = —fa2 — 4ABe2^, fa' = 1 — 2fa.

In the case k = 0 one can integrate the corresponding dynamical system and express the solution explicitly

1 C pt

TTcl +(faciy2W + Be-x),

where Ci,C2,A,B are arbitrary constants, while AB = 0.

Example 2. Consider the nonlinear heat equation

/ \ 2 2 2

Uf = (UUx)x + ^ xx + ^x + ^ .

The special solution in the form (8) yields the following equation

fa + V = (fa + V )e* V” + e2^ F'2 + fa2 + 2e* V + e2^ V2.

Splitting the last equation yields

fa = fa2 + e2^ (VV" + V '2 + V2), V = faV" + 2V.

Then we have the following overdetermined system:

W’ ’ + V '2 + V2 = k = const, V" = mV, m = const.

From the last relations we have

V (x) = —2C cos —=, C = const.

V2

So, we can write down the family of solution as

u(t, x) = fa(t) + e^—2C cos —=,

where functions fa(t)/fa(t) satisfy the following dynamical system:

fa = fa2 + C2e2^, = 2 - fa.

4. Some generalizations within linear superposition

1. The linear superposition evolution can easily be generalized for higher order timederivatives. Indeed, one can prolong evolutionary system (6) for partial derivative utt,uttt,...:

utt(t,x) = fa(t) + fa(t)4>'(t) + fa (t) + (t))u(t,x),... .

In that case the solution has evidently the same form

u(t,x) = fa(t) + e^(t')V (x). (9)

To determine unknown functions fa(t),tf(t),V(x) one should substitute (9) into corresponding evolutionary equation.

2. The linear superposition evolution can be generalized for 2-D space-dimensional solutions u(t,x,y). For that case we prolong the evolution system (4) for partial derivative

, Uy, Uxx, ^yy, —

ut(t, x, y) = fa(t) + (t)u(t, x, y), (10)

Uxt(t,X,y) = (t)ux(t,X,y), Uyt(t,X,y) = (t)Uy (t,x,y),

Uxxt(t,X,y)= (t)uxx(t,X,y), Uyyt(t,X,y)= (t)Uyy(t,X,y),. . . .

In 2-D case the solution has similar to 1-D case form

u(t,x) = fa(t) + e^V (x,y),

and the same linear superposition formula.

Below we consider an example of evolutionary equation of the second order

Ut F fa ux, Uy, uxx, Uyy, uXy),

which is compatible with the evolution (10).

Example3. Let us consider the nonlinear equation

Ut U(UXX + Uyy^ + 2UxxUyy Ux Uy . (11)

We substitute a solution

u(t,x) = fa(t) + e^(t')V (x,y),

and obtain

fa + V = (fa + Ve*)(VXX + Vyy)e* + 2e2^VxxVyy - e2^(V2 + V2).

Then we split the last equality

V {Vxx + Vyy) + WxxVyy — (Vx2 + Vy2) = k, k = const.,

mV = Vxx + Vyy, m = const, fa = ke2^, mrfa = fa.

For the special case m = —1,k = —2 one can find the special solution

V (x, y) = cos x + cos y, and then write the solution for equation (11) in the form

u(t,x,y) = fa(t) + (cos x + cos y), while fa(t),fa(t) should obey the following dynamical system

fa = —2e2^, fa' = — fa,

which can be integrated completely.

3. The linear superposition evolution can be extended for four-dimensional Lie Algebra, involving variable x as non-evolutionary parameter.

Indeed, let us consider the following extended evolution system (4):

Ut(t,x) = fai(t) + fa2(t)u + fa;i(t)x + fa4(t)x2, (12)

Uxt(t,x) = fa2 (t)Ux + fa3(t) + 2faA(t)x, Uxxt(t,x) = fa2 (t)Uxx + 2faA(t), ....

which still has linear superposition for its solutions. Thus, formally we have two dependent

dx

variables u,x, while the second one does not evolute in time: — = 0, since x = const. The

dt

corresponding Lie Algebra is the following four-dimensional algebra L4, which we prolong for high-order derivatives:

v ^ v d d 5 d d 2 d d d

Xi = —, X2 = u—---+ U^— + Uxx, X3 = x——, X4 = X —----\-2X— +2~ .

UU UU UU x OU xx OU OU x OU OU x OU xx

In accordance with the Lie theorem we have two dependent variables u,x since n = 2, and four-dimensional algebra L4, consequently r = 4. The Lie theorem yields m > 2. In fact two particular solutions cast the minimum number of solutions and allow to write down a general solution of equations (12). Thus, (12) has the fundamental set of special solutions ui and u2 with superposition

U — Ui _ ^

^ ,

U2 — Ui

and a general solution is u = (1 — C)ui + Cu2, C = const. In accordance with the evolution (12) a special solution has the following form:

u(t,x) = fai(t) + fa2(t)V (x) + fa3(t)x + fa 4 (t)x2, (13)

where functions fai(t) are connected with the functions faj (t) by means of the following dynamical system

fa'i = fai + fa2fai, fa2 = fa2fa2, fa3 = fa2fa3 + fa3, fa4 = fa2fa4 + fa 4.

Thus, we got the new representation of solution, which possesses two additional functions of t. Now we consider the example of such extension.

Example 4. We consider the nonlinear equation

Ut = uxx (14)

and seek the special solution of equation (14) in the form (13). It yields the following equation fai + fa'2y + fa3x + fa'4x2 = fa22(V'')2 V + 4fa42 + 4fa4fa2V''.

Splitting the last equality and integrating corresponding equations for V(x) and fai(t) yields u(t,x) = C5 - C(Ci - l44t)1/3 + C14x + (X + ~C[x + ~\6) +

i / Q/^2 ri3 '

+Cy - nu (/r‘+02x3 + ~nlx2 + il1+ Û

where C1,C2,C3,C4, C5 are arbitrary constants.

Thus, we developed the 4-order in x polynomial expression for the solution, which contains three independent functions of t.

5. The subset of solutions with nonlinear superposition principle We now consider the second subalgebra L2 spanned by the operators:

O O

Xi = U — , X2 = u2 — , [Xl,X2]=X2. (15)

The prolongation of the L2 for spatial derivatives

O O O 2 O O 2 O

X1 = U^ + Ux^~ + Uxx^~ , X2 = ii — +2uux— + 2(uuxx + u,x )— ,

OU OU x OU xx OU OU x OU xx

corresponds the following evolution (the Bernoulli equation):

ut = <f)(t)u + fa(t)u2, (16)

Uxt <{t)ux + fa{t)2wix, Uxxt <(t)uxx + fa{t)2(wixx + U x).

Let apply the point change of variable

1

u(t, x) = —

v(t, x)

O O

then the subalgebra (15) becomes Y1 = —v—, Y2 = — ; [1^,121 = Y2. The corresponding nonO O

autonomous evolution (16) is transforming into the linear one

vt = — < (t)v + fa(t).

Thus, now we have a linear superposition situation, which defines the family of solutions in the form

v(t ,x) = fa(t) + e-^(t^a(x).

Coming back to u(t,x), we have the solutions in the form

= —1

, fait) + e-^(t)a(x),

which possess the following nonlinear superposition principle for its solutions:

U(t x) =____________ww______ (17)

w ,x) CU2 + (1 — C)wi. ( )

One can apply that approach to the Example l and obtain the following nonlinear equations with nonlinear superposition (17):

nwx 2 ux2

Ut = Uxx — 2-------\ 2----1.

U U2

6. The subset of solutions with the Riccati-type nonlinear superposition We now consider the complete Riccati equation, prolonged to evolution of space derivatives

ut = a(t) + 3 (t)u + y(t)u2, (18)

uxt 3 (t)ux + 2y(t)uux, uxxt P(t)uxx + 2^(t)(uuxx + UX) '

Now we will construct a solution of the Riccati equation. Let u1 be a particular solution of the Riccati equation (18). Then one can change variables as follows:

u(t, x) = u1 + w(t, x),

and get the corresponding Bernoulli’s equation and its differential sequences

wt = (3 + 2yui)w + yw2, wxt = (3 + 2yui)wx + 2ywwx,

wxxt = (3 + 2y ui)wxx + 2y(wwxx + w2x).

The change

\

w(t, x) = --

v(t, x)

yields the following linear equation

vt = y(t)+fa it) v, fa = —(/3 + 2jwi).

The last equation possesses a linear superposition principle and has general solution

v(t ,x) = <(t) + e'^C (x), where <(t) is a particular solution of a linear equation:

< = 7 + fa<.

The corresponding solution of the Bernoulli equation

1

w(t, x) = —-

4>(t) + e^(t)C (x) yields the solution of the Riccati system

1

u( t, x) = —

fa(t) + e^(t)C(x) ’

which possesses the nonlinear superposition of the particular solutions.

In the following example we develop an equation which possesses solutions with the Riccati-type superposition.

2 2

Example 5. Let consider a special integrable case of the Riccati equation ut = -z — u ,

t2

which has a general solution

3t2 1

u(x, f) = w—ciX) — ï <19)

Applying a shift by special solution u0 = —1/t we transform the Ruccati equation into the Bernoulli equation

2 2 1

vt = -v — V , u = v----------.

t ’ t

We linearize the Bernoulli equation by change w = —

2

wt = — jw — 1, <20)

and find the solution

C( x)

w(x, t)

t2 3'

The evolution (20) is compatible with the following equation

1

wt \jwxwxx 3 .

The substitution of solution in form (19) yields the equation

(CC )5/2 + -C2 = c1, c1 = cons t;

which can be solved in quadrature for C(x):

f dC x =J (d - IC*^ + C2' C'2 = ^

Applying backward transformation we obtain the following equation for v(x, t):

vxx nvl\ v2

vt = V^[---------2^2 -—,

\ v v2 J 3

and then the evolutionary equation for u( x, ):

ut = yUx ; 2

tuxx u2x t2 \ tu2 + 2u

1 + tu (1 + tu)2 J 3t

The last equation has solution (19) and Riccati-type superposition for particular solutions.

7. How to separate evolutionary equations which possess a linear subspace

of solutions?

We now consider the compatibility conditions of the evolution (6) with a PDE

ut — F (u,ux,uxx) (21)

as a compatibility with certain differential constraint. There is a lot of different approaches to constructing exact solutions for PDEs based on differential constraints (see, for example, [14] and references therein). We restrict ourselves here with such constraint, which leads to solutions with linear superposition principle.

Excluding two functions of t by differentiation we rewrite linear evolution equations (6) as differential constrain

uxxtux uxtuxx. (22)

Thus, now we can repose the problem of compatibility as a compatibility of (21) and differential constrain (22). We substitute time derivatives from (21) into (22) and obtain ODE

D, () =0.

\ ux J

In the same way one can write down the constrain of k-th order:

Drl D(F1)=q, k=1,2,.... (23)

uk

The operator of total differentiation with respect to t along trajectories of (21)

8 8

X * = F (u,ux,uxx)— + D2X (F)— + ...,

8u 8u xx

casts a higher order symmetry operator. Then the criterion of an invariance of manifold (23) reads

'Dx(F )'

X * D

(dx (-

= 0. (24)

(23)

Being solved for F equation (24) yields evolution equation which potentially has linear subspace of solutions. Existence and particular form of solutions can be obtained by substituting solution into evolution equation and applying splitting procedure, which was demonstrated by examples.

In a similar way one can write down the compatibility condition for evolutionary PDEs which possesses a subset of solutions with Riccati-type superposition.

Concluding remarks

Thus, we considered the Lie non-autonomous dynamical equations (NADS) as some sort of external action on a given evolutionary equation. It makes it possible to find a subset of special solutions of evolutionary equation, which possesses a superposition principle which acts within subset of solutions. This leads to integration of ordinary differential equations in a process of constructing exact solutions of PDEs. In this paper we considered the application of the most simple one-dimensional case of the Lie theorem. It also was shown that the NADS approach can be generalized for 1+2 D equations as well. Feather generalizations will be published elsewhere.

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Vladimir A. Dorodnitsyn,

Keldysh Institute of Applied Mathematics RAS,

Miusskaya sq. 4,

125047 Moscow, Russia

E-mail: [email protected]