First Order Differential Constraints

S.8.2. First Order Differential Constraints

S.8.2-1. Second-order evolution equations. Consider a general second-order evolution equation solved for the highest-order derivative:

* In the general case, for the investigation of overdetermined systems one should utilize methods based on: (i) the Cartan algorithm or (ii) the Janet–Spenser–Kuranishi algorithm. A description of these algorithms and other relevant information regarding the theory of overdetermined systems can be found, for instance, in the works of M. Kuranishi (1967), J. F. Pommaret (1978), A. F. Sidorov, V. P. Shapeev, and N. N. Yanenko (1984).

Original equation: Fxyww (,, , x , ww y , xx , w xy , w yy , ...) = 0

Introduce a supplementary equation

Differential constraint: Gxywwww (,, , x , y , xx , w xy , w yy , ...) = 0

Perform compatibility analysis of the two equations

Find compatibility conditions for the equations F = 0 and G =0

Obtain equations for the determining functions

Solve the equations for the determining functions

Insert the solution into the differential constraint

Find an invariant manifold: gxywwww (,, , x , y , xx , w xy , w yy , ...) = 0

Solve the equation g = 0 for w

Insert resulting solution (with arbitrariness) into original equation

Determine the unknown functions and constants

Obtain an exact solution of the original equation

Figure 5. Algorithm for the construction of exact solutions by the differential constraints method

Let us supplement this equation with a first-order differential constraint

The condition of compatibility of these equations is obtained by differentiating (14) with respect to t once and differentiating (15) with respect to x twice, and then equating the two resulting expressions for the third derivatives w xxt :

D t 2 F=D x G. (16) Here, D t and D x are the total differentiation operators with respect to t and x: ∂

+ w xt . (17) ∂t

∂w x ∂w t The partial derivatives w t , w xx , w xt , and w tt in (17) should be expressed in terms of x, t, w, and w x by means of the relations (14), (15) and those obtained by differentiation of (14), (15). As a result,

∂w x +F ∂w x ∂w x In the expression for

F, the derivative w t should be replaced by

G by virtue of (15).

Example 2. From the class of nonlinear heat equations with a source

∂w = ∂

f (w) ∂w + g(w),

∂t

∂x

∂x

∂w = ϕ(w).

∂t

Equations (19) and (20) are special cases of (14) and (15) with

w t − f ′ ( w)w 2 x − g(w)

ϕ(w) − g(w) − f ′ ( w)w 2 x

G = ϕ(w). The functions

F=

= f (w) , f (w)

f (w), g(w), and ϕ(w) are unknown in advance and are to be determined in the subsequent analysis. Using (18) and (17), we find partial derivatives and the total differentiation operators:

We insert the expressions of D x and D t into the compatibility conditions (16) and rearrange terms to obtain

In order to ensure that this equality holds true for any w x , one should take

Nondegenerate case . Assuming that the function f = f (w) is given, we obtain a three-parameter solution of equations (21) for the functions g(w) and ϕ(w):

g(w) =

a + cf

f dw + b ,

ϕ(w) = a f dw + b ,

where a, b, and c are arbitrary constants. We substitute ϕ(w) of (22) into equation (20) and integrate to obtain

f dw = θ(x)e at − b. ( 23 )

Differentiating (23) with respect to x and t, we get w t = ae at θ/f and w x = e at θ x ′ /f . Substituting these expressions

into (19) and taking into account (22), we arrive at the equation θ xx ′′ + cθ = 0, whose general solution is given by

C 1 sinh x√−c + C 2 cosh x√−c

where C 1 and C 2 are arbitrary constants. Formulas (23)–(24) describe exact solutions (in implicit form) of equation (19) with

f (w) arbitrary and g(w) given by (22). Degenerate case . There also exists a two-parameter solution of equations (21) for the functions g(w) and ϕ(w) (as above,

f is assumed arbitrary):

g(w) = b + c,

ϕ(w) = b ,

where b and c are arbitrary constants. This solution can be obtained from (22) by renaming variables, b → b/a and c → ac/b, and letting

a → 0. After simple calculations, we obtain the corresponding solution of equation (19) in implicit form:

f dw = bt − cx + C 1 x+C 2 .

Reference : V. A. Galaktionov (1994).

The example given below shows that calculations may be performed without the use of the general formulas (16)–(18).

Example 3. Consider the problem of finding second-order nonlinear equations

admitting first-order invariant manifolds of the form

∂w = 1 (

∂w

g w)

+ g 0 ( w).

∂t

∂x

The functions f 1 ( w), f 0 ( w), g 1 ( w), and g 0 ( w) are unknown in advance and are to be determined in the subsequent analysis. First, we calculate derivatives. Equating the right-hand sides of (25) and (26), we get

w xx = h 1 w x + h 0 ,

where h 1 = g 1 − f 1 , h 0 = g 0 − f 0 .

Here and in what follows, the argument of the functions f 1 , f 0 , g 1 , g 0 , h 1 , and h 0 is omitted. Differentiating (26) with respect to x twice and using the expression (27) for w xx , we find the mixed derivatives

w 2 xt = g 1 w 2 xx + g ′ 1 w x + g ′ 0 w x = g ′ 1 w x +( g 1 h 1 + g ′ 0 ) w x + g 1 h 0 ,

w xxt = g 1 w x +( g 1 h 1 +3 g 1 h 1 + g 0 ) w x +( g 1 h 0 ′ +3 g 1 ′ 0 + 1 h 2 g h 1 + g 0 ′ h 1 ) w x +( g 1 h 1 + g ′ 0 ) h 0 , where the prime denotes a derivative with respect to w. Differentiating (27) with respect to t and using the expressions (26)

and (28) for w t and w xt , we obtain w 2 xxt = h 1 w xt + h 1 ′ w x w t + h ′ 0 w t =( g 1 h ′ 1 + g ′ 1 h 1 ) w x +( g 1 h 1 2 + g ′ 0 h 1 + g 0 h ′ 1 + g 1 h ′ 0 ) w x + g 1 h 0 h 1 + g 0 h ′ 0 .

We equate the expressions for the third derivative w xxt from (28) and (29) and collect terms with the same power of w x to obtain an invariance condition in the form

g ′′ 1 w 3 x + (2 g ′ 1 h 1 + g ′′ 0 ) 2 w x + (3 g ′ 1 h 0 − g 0 h ′ 1 ) w x + g ′ 0 h 0 − g 0 h ′ 0 = 0.

For condition (30) to hold we require that the coefficients of like powers of w x

be zero:

g 0 ′ h 0 − g 0 h ′ 0 = 0. The general solution of this system of ordinary differential equations is given by the following formulas:

where C 1 , ...,C 6 are arbitrary constants. Using formulas (27) for h 0 and h 1 together with (31), we find the unknown functions involved in equations (25) and (26):

Let us dwell on the special case of

C 6 = bk in (32), where

C 1 =− k, C 2 = C 4 = 0, C 3 = −1 /k,

C 5 = ak,

a, b, and k are arbitrary constants (k ≠ 0). The corresponding equation (25) and the invariant manifold (26) have the form

w t = w xx −( k + 3)ww x +( k + 1)(w 3 + aw + b),

The general solution of the first-order quasilinear equation (34) can be written out in implicit form; it involves the integral I(w) = R w(w 3 + aw + b) −1 dw and its inversion. Due to its complex structure, this solution is inconvenient for the construction of exact solutions of equation (33). In this situation, instead of (34) one can use equations obtained from (33) and (34) by eliminating the derivative w t :

This ordinary differential equation coincides with (27), where h 1 and h 0 are expressed by (31). The substitution w = −U x /U transforms (35) into a third-order linear equation with constant coefficients,

whose solutions are determined by the roots of the cubic equation λ 3 + aλ − b = 0. In particular, if all its roots λ n are real, then the general solutions of equations (35) and (36) are given by

w = −U x /U ,

U=r 1 ( t) exp(λ 1 x) + r 2 ( t) exp(λ 2 x) + r 3 ( t) exp(λ 3 x).

The functions r n ( t) are found by substituting (37) into equation (33) or (34). Note that equation (33) was studied in more detail by another method in Subsection S.6.3 (see Example 7 with a=1 and b 2 = 0).

Remark 1. In the general case, for a given function F, the compatibility condition (16) is

G. This equation admits infinitely many solutions (by the theorem about the local existence of solutions). Therefore, the second-order partial

a nonlinear partial differential equation for the function

differential equation (14) admits infinitely many compatible first-order differential constraints (15). Remark 2. In the general case, the solution of the first-order partial differential equation (15)

reduces to the solution of a system of ordinary differential equations; see Kamke (1965) and Polyanin, Zaitsev, and Moussiaux (2002).

S.8.2-2. Second-order hyperbolic equations. In a similar way, one can consider second-order hyperbolic equations of the form

supplemented by a first-order differential constraint (15). Assume that G w x ≠ 0. The compatibility condition for these equations is obtained by differentiating (38) with respect to t and (15) with respect to t and x, and then equating the resulting expressions of the third derivative w xtt to one another:

(39) Here, D t and D x are the total differential operators of (17) in which the partial derivatives

D t F=D x [D t G].

w t , w xx , w xt , and w tt must be expressed in terms of x, t, w, and w x with the help of relations (38) and (15) and those obtained by differentiating (38) and (15).

Let us show how the second derivatives can be calculated. We differentiate (15) with respect to x and replace the mixed derivative by the right-hand side of (38) to obtain the following expression for the second derivative with respect to x:

=H 1 x, t, w, . (40) ∂x

∂x 2 ∂x Here and in what follows, we have taken into account that (15) allows us to express the derivative

with respect to t through the derivative with respect to x. Further, differentiating (15) with respect to t yields

=H 2 x, t, w, . (41) ∂t

∂t 2 ∂x Replacing the derivatives w t , w xt , w xx , and w tt in (17) by their expressions from (15), (38), (40),

and (41), we find the total differential operators D t and D x , which are required for the compatibility condition (39).

Example 4. Consider the nonlinear equation

with two different first-order differential constraints. Case 1. Let us supplement (42) with a quasilinear differential constraint of the form

∂w = g(w) ∂w .

∂t

∂x

Simple calculations combined with the compatibility conditions (39), where F = f(w) and G = g(w)w x , lead us to the expression

3 fg ′ w x +[ gg ′′ −( g ′ ) 2 ] w 3 x = 0.

Equating the coefficients of like powers of w x to zero, we find that g = const. This corresponds to a traveling-wave solution of equation (42), w = w(kx + λt).

Case 2. Now let us supplement equation (42) by a differential constraint with a quadratic nonlinearity in derivatives,

∂w ∂w =

g(w).

∂t ∂x

Calculations with the help of the compatibility condition (39), where F = f(w) and G = g(w)/w x , lead us to an expression relating the functions

f = f (w) and g = g(w):

It can be shown that the differential constraint (43), together with the compatibility condition (44), yields a self-similar

solution w = w(xt) of equation (42); here, x and t can be replaced by x + C 1 and t+C 2 .

S.8.2-3. Second-order equations of general form. Consider a second-order hyperbolic equation of the general form

(45) with a first-order differential constraint

F(x, t, w, w x , w t , w xx , w xt , w tt )=0

(46) Let us successively differentiate equations (38) and (39) with respect to both variables so as to

G(x, t, w, w x , w t ) = 0.

obtain differential relations involving second and third derivatives. We get

G = 0, D x [D x G] = 0, D x [D t G] = 0, D t [D t G] = 0. (47) The compatibility condition for (45) and (46) can be found by eliminating the derivatives w t , w xx ,

w xt , w tt , w xxx , w xxt , w xtt , and w ttt from the nine equations of (45)–(47). In doing so, we obtain an expression of the form

(48) If the left-hand side of (48) is a polynomial in w x , then the compatibility conditions result from ❆✂❇ equating the functional coefficients of the polynomial to zero.

H(x, t, w, w x ) = 0.

References for Subsection S.8.2: A. F. Sidorov, V. P. Shapeev, and N. N. Yanenko (1984), V. A. Galaktionov (1994), P. J. Olver (1994), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Rodionov (1999).