The procedure of the construction of exact solutions to nonlinear equations of mathematical physics by the differential constraints method consists of several steps described below.. 15.
Trang 1Substituting the derivatives of the function w from (15.10.1.5) and (15.10.1.6) into (15.10.1.7), we obtain the following third-order ordinary differential equation for ϕ:
(ϕ y)2– ϕϕ yy = aϕ yyy, (15 10 1 8 ) which represents the compatibility condition for equations (15.10.1.4) and (15.10.1.5).
In order to construct an exact solution, we integrate equation (15.10.1.5) to obtain
w = ϕ(y)x + ψ(y). (15 10 1 9 )
The function ψ(y) is found by substituting (15.10.1.9) into (15.10.1.4) and taking into account the
condi-tion (15.10.1.8) As a result, we arrive at the ordinary differential equacondi-tion
ϕ y ψ y – ϕψ yy = aψ yyy (15 10 1 10 )
Finally, we obtain an exact solution of the form (15.10.1.9), with the functions ϕ and ψ described by equations
(15.10.1.8) and (15.10.1.10).
Remark It is easier to obtain the above solution by directly substituting expression (15.10.1.9) into the original equation (15.10.1.4).
15.10.1-2 General description of the differential constraints method
The procedure of the construction of exact solutions to nonlinear equations of mathematical physics by the differential constraints method consists of several steps described below
1◦ In the general case, the identification of particular solutions of the equation
F
x, y, w, ∂w
∂x,∂w
∂y,∂2w
∂x2,
∂2w
∂x∂y,∂2w
∂y2 ,
is performed by supplementing this equation with an additional differential constraint
G
x, y, w, ∂w
∂x,∂w
∂y,∂
2w
∂x2,
∂2w
∂x∂y,∂
2w
∂y2 ,
=0 (15.10.1.12)
The form of the differential constraint (15.10.1.12) may be prescribed on the basis
of (i) a priori considerations (for instance, it may be required that the constraint should represent a solvable equation); (ii) certain properties of the equation under consideration (for instance, it may be required that the constraint should follow from symmetries of the equation or the corresponding conservation laws)
2◦ In general, the thus obtained overdetermined system (15.10.1.11)–(15.10.1.12) requires
a compatibility analysis If the differential constraint (15.10.1.12) is specified on the ba-sis of a priori considerations, it should allow for sufficient freedom in choosing func-tions (i.e., involve arbitrary determining funcfunc-tions) The compatibility analysis of system (15.10.1.11)–(15.10.1.12) should provide conditions that specify the structure of the
deter-mining functions These conditions (compatibility conditions) are written as a system of
ordinary differential equations (or a system of partial differential equations)
Usually, the compatibility analysis is performed by means of differentiating (possibly,
several times) equations (15.10.1.11) and (15.10.1.12) with respect to x and y and
elimi-nating the highest-order derivatives from the resulting differential relations and equations (15.10.1.11) and (15.10.1.12) As a result, one arrives at an equation involving powers of lower-order derivatives Equating the coefficients of all powers of the derivatives to zero, one obtains compatibility conditions connecting the functional coefficients of equations (15.10.1.11) and (15.10.1.12)
Trang 23◦ One solves the system of differential equations obtained in Item2◦for the determining
functions Then these functions are substituted into the differential constraint (15.10.1.12)
to obtain an equation of the form
g
x, y, w, ∂w
∂x, ∂w
∂y,∂
2w
∂x2,
∂2w
∂x∂y,∂
2w
∂y2,
=0 (15.10.1.13)
A differential constraint (15.10.1.13) that is compatible with equation (15.10.1.11) under
consideration is called an invariant manifold of equation (15.10.1.11).
4◦ One should find the general solution of (i) equation (15.10.1.13) or (ii) some equation
that follows from equations (15.10.1.11) and (15.10.1.13) The solution thus obtained will involve some arbitrary functions{ϕ m}(these may depend on x and y, as well as w) Note
that in some cases, one can use, instead of the general solution, some particular solutions
of equation (15.10.1.13) or equations that follow from (15.10.1.13)
5◦ The solution obtained in Item 4◦ should be substituted into the original equation
(15.10.1.11) As a result, one arrives at a functional differential equation from which the functions {ϕ m} should be found Having found the {ϕ m}, one should insert these functions into the solution from Item4◦ Thus, one obtains an exact solution of the original
equation (15.10.1.11)
Remark 1 Should the choice of a differential constraint be inadequate, equations (15.10.1.11) and (15.10.1.12) may happen to be incompatible (having no common solutions).
Remark 2 There may be several differential constraints of the form (15.10.1.12).
Remark 3 At the last three steps of the differential constraints method, one has to solve various equations (systems of equations) If no solution can be constructed at one of those steps, one fails to construct an exact solution of the original equation.
For the sake of clarity, the general scheme of the differential constraints method is represented in Figure 15.6
15.10.2 First-Order Differential Constraints
15.10.2-1 Second-order evolution equations
Consider a general second-order evolution equation solved for the highest-order derivative:
∂2w
∂x2 =F
x, t, w, ∂w
∂x,∂w
∂t
Let us supplement this equation with a first-order differential constraint
∂w
∂t =G
x, t, w, ∂w
∂x
The condition of compatibility of these equations is obtained by differentiating (15.10.2.1)
with respect to t once and differentiating (15.10.2.2) with respect to x twice, and then equating the two resulting expressions for the third derivatives w xxt and w txx:
Dt F = D2x G. (15.10.2.3)
Trang 3Introduce a supplementary equation
Perform compatibility analysis of the two equations
Obtain equations for the determining functions
Insert the solution into the differential constraint
Determine the unknown functions and constants
Solve the equations for the determining functions
Insert resulting solution (with arbitrariness) into the original equation
Obtain an exact solution of the original equation
Figure 15.6 Algorithm for the construction of exact solutions by the differential constraints method.
Here, Dtand Dx are the total differentiation operators with respect to t and x:
Dt= ∂
∂t + w t ∂
∂w + w xt ∂
∂w x + w tt
∂
∂w t, Dx =
∂
∂x + w x ∂
∂w + w xx ∂
∂w x + w xt
∂
∂w t.
(15.10.2.4)
The partial derivatives w t , w xx , w xt , and w tt in (15.10.2.4) should be expressed in terms
of x, t, w, and w xby means of the relations (15.10.2.1), (15.10.2.2) and those obtained by differentiation of (15.10.2.1), (15.10.2.2) As a result, we get
w t=G, w xx =F, w xt= Dx G = ∂ G
∂x + w x ∂ G
∂w +F ∂ G
∂w x,
w tt= Dt G = ∂ G
∂t +G ∂ G
∂w + w xt ∂ G
∂w x =
∂ G
∂t +G ∂ G
∂w +
∂ G
∂x + w x ∂ G
∂w +F ∂ G
∂w x
∂ G
∂w x.
(15.10.2.5)
In the expression forF, the derivative w tshould be replaced byG by virtue of (15.10.2.2).
Example 1 From the class of nonlinear heat equations with a source
∂w
∂t = ∂
∂x
f (w) ∂w
∂x
+ g(w), (15 10 2 6 )
Trang 4let us single out equations possessing invariant manifolds of the simplest form
∂w
∂t = ϕ(w). (15 10 2 7 ) Equations (15.10.2.6) and (15.10.2.7) are special cases of (15.10.2.1) and (15.10.2.2) with
F = w t – f (w)w2x – g(w)
ϕ (w) – g(w) – f (w)w2x
The functions f (w), g(w), and ϕ(w) are unknown in advance and are to be determined in the subsequent
analysis.
Using (15.10.2.5) and (15.10.2.4), we find partial derivatives and the total differentiation operators:
w t = ϕ, w xx=F, wxt = ϕ w x, w tt = ϕϕ ,
Dt= ∂
∂t + ϕ ∂
∂w + ϕ w x ∂
∂w x + ϕϕ ∂
∂w t, Dx= ∂
∂x + w x ∂
∂w +F ∂
∂w x + ϕ w x ∂
∂w t.
We insert the expressions of Dx and Dtinto the compatibility conditions (15.10.2.3) and rearrange terms to
(f ϕ)
f
w2x+ϕ – g
– ϕ
ϕ – g
f
= 0
In order to ensure that this equality holds true for any w x, one should take
(f ϕ)
f
= 0 , ϕ – g
– ϕ
ϕ – g
f
= 0 (15 10 2 8 )
Nondegenerate case Assuming that the function f = f (w) is given, we obtain a three-parameter solution
of equations (15.10.2.8) for the functions g(w) and ϕ(w):
g (w) = a + cf
f
f dw + b
, ϕ (w) = a
f
f dw + b
, (15 10 2 9 )
where a, b, and c are arbitrary constants.
We substitute ϕ(w) of (15.10.2.9) into equation (15.10.2.7) and integrate to obtain
f dw = θ(x)e at – b. (15 10 2 10 )
Differentiating (15.10.2.10) 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 (15.10.2.6) and taking into account (15.10.2.9), we arrive at the equation θ xx + cθ =0 , whose general solution is given by
θ=
⎧
⎨
⎩
C1sin x √
c
+ C2cos x √
c
if c >0 ,
C1sinh x √ –c + C
2 cosh x √ –c if c <0,
C1x + C2 if c =0 ,
(15 10 2 11 )
where C1 and C2 are arbitrary constants Formulas (15.10.2.10)–(15.10.2.11) describe exact solutions (in
implicit form) of equation (15.10.2.6) with f (w) arbitrary and g(w) given by (15.10.2.9).
Degenerate case There also exists a two-parameter solution of equations (15.10.2.8) for the functions
g (w) and ϕ(w) (as above, f is assumed arbitrary):
g (w) = b
f + c, ϕ (w) = b
f,
where b and c are arbitrary constants This solution can be obtained from (15.10.2.9) by renaming variables,
b → b/a and c → ac/b, and letting a →0 After simple calculations, we obtain the corresponding solution of equation (15.10.2.6) in implicit form:
f dw = bt – 1
2cx2+ C1x + C2.
The example given below shows that calculations may be performed without the use of the general formulas (15.10.2.3)–(15.10.2.5).
Remark 1 In the general case, for a given functionF, the compatibility condition (15.10.2.3) is a
non-linear partial differential equation for the functionG This equation admits infinitely many solutions (by
the theorem about the local existence of solutions) Therefore, the second-order partial differential equa-tion (15.10.2.1) admits infinitely many compatible first-order differential constraints (15.10.2.2).
Remark 2 In the general case, the solution of the first-order partial differential equation (15.10.2.2) reduces to the solution of a system of ordinary differential equations; see Chapter 13.
Trang 515.10.2-2 Second-order hyperbolic equations.
In a similar way, one can consider second-order hyperbolic equations of the form
∂2w
∂x∂t =F
x, t, w, ∂w
∂x,∂w
∂t
supplemented by a first-order differential constraint (15.10.2.2) Assume thatG w x ≠ 0
A compatibility condition for these equations is obtained by differentiating (15.10.2.12)
with respect to t and (15.10.2.2) with respect to t and x, and then equating the resulting expressions of the third derivatives w xtt and w ttx to one another:
Dt F = D x[Dt G]. (15.10.2.13) Here, Dt and Dx are the total differential operators of (15.10.2.4) in which the partial
derivatives 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 (15.10.2.12) and (15.10.2.2) and those obtained by differentiating (15.10.2.12) and (15.10.2.2)
Let us show how the second derivatives can be calculated We differentiate (15.10.2.2)
with respect to x and replace the mixed derivative by the right-hand side of (15.10.2.12) to obtain the following expression for the second derivative with respect to x:
∂ G
∂x + w x ∂ G
∂w + w xx ∂ G
∂w x =F
x, t, w, ∂w
∂x, ∂w
∂t
=⇒ ∂2w
∂x2 =H1
x, t, w, ∂w
∂x
(15.10.2.14) Here and in what follows, we have taken into account that (15.10.2.2) allows us to
ex-press the derivative with respect to t through the derivative with respect to x Further, differentiating (15.10.2.2) with respect to t yields
∂2w
∂t2 =
∂ G
∂t +w t ∂ G
∂w +w xt ∂ G
∂w x =
∂ G
∂t +G ∂ G
∂w+F ∂ G
∂w x =⇒ ∂2w
∂t2 =H2
x, t, w, ∂w
∂x
(15.10.2.15)
Replacing the derivatives w t , w xt , w xx , and w tt in (15.10.2.4) by their expressions from (15.10.2.2), (15.10.2.12), (15.10.2.14), and (15.10.2.15), we find the total differential operators Dtand Dx, which are required for the compatibility condition (15.10.2.13)
Example 2 Consider the nonlinear equation
∂2w
∂x∂t = f (w). (15 10 2 16 ) Let us supplement (15.10.2.16) with quasilinear differential constraint of the form
∂w
∂x = ϕ(t)g(w). (15 10 2 17 )
Differentiating (15.10.2.16) with respect to x and then replacing the first derivative with respect to x by
the right-hand side of (15.10.2.17), we get
w xxt = ϕgf w (15 10 2 18 )
Differentiating further (15.10.2.17) with respect to x and t, we obtain two relations
w xx = ϕg w w x = ϕ2gg w , (15 10 2 19 )
w xt = ϕ t g + ϕg w w t (15 10 2 20 ) Eliminating the mixed derivative from (15.10.2.20) using equation (15.10.2.16), we find the first derivative with
respect to t:
w t= f – ϕ
t g
Trang 6Differentiating (15.10.2.19) with respect to t and replacing w tby the right-hand side of (15.10.2.21), we have
w xxt= 2ϕϕ
t gg w + ϕ2(gg w) w w t= 2ϕϕ
t gg w + ϕ(gg w) w f – ϕ
t g
g w (15 10 2 22 )
Equating now the third derivatives (15.10.2.18) and (15.10.2.22), canceling them by ϕ, and performing simple
rearrangements, we get the determining equation
ϕ t g [(g w )2– gg ww ] = gg w f w – f (gg w ) w, (15 10 2 23 ) which has two different solutions.
Solution 1 Equation (15.10.2.23) is satisfied identically for any ϕ = ϕ(t) if
(g w)2– gg ww= 0 ,
gg w f w – f (gg w) w= 0 The general solution of this system has the form
f (w) = ae λw, g (w) = be λw/2, (15 10 2 24 )
where a, b, and λ are arbitrary constants For simplicity, we set
a = b =1 , λ= – 2 (15 10 2 25 )
Substitute g(w) defined by (15.10.2.24)–(15.10.2.25) into the differential constraint (15.10.2.17) and
integrate the resulting equation to obtain
w = ln[ϕ(t)x + ψ(t)], (15 10 2 26 )
where ψ(t) is an arbitrary function Substituting (15.10.2.26) into the original equation (15.10.2.16) with the right-hand side (15.10.2.24)–(15.10.2.25), we arrive at a linear ordinary differential equation for ψ(t):
ψϕ t – ϕψ t= 1 The general solution of this equation is expressed as
ψ (t) = Cϕ(t) – ϕ(t)
dt
where C is an arbitrary constant.
Thus, formulas (15.10.2.26)–(15.10.2.27), where ϕ(t) is an arbitrary function, define an exact solution to the nonlinear equation w xt = e– 2w
Solution 2 The second solution is determined by the linear relation
ϕ (t) = at + b, (15 10 2 28 )
where a and b are arbitrary constants In this case, the functions f (w) and g(w) are related by (15.10.2.23), with ϕ t = a Integrating (15.10.2.17) with constraint (15.10.2.28) yields the solution structure
w = w(z), z = (at + b)x + ψ(t), (15 10 2 29 )
where ψ(t) is an arbitrary function Inserting it into the original equation (15.10.2.16) and changing x to z with
the help of (15.10.2.29), we obtain
[az + (at + b)ψ t – aψ]w zz + aw z = f (w). (15 10 2 30 )
In order for this relation to be an ordinary differential equation for w = w(z), one should set
(at + b)ψ t – aψ = const.
Integrating yields ψ(t) in the form
ψ (t) = ct + d, (15 10 2 31 )
where c and d are arbitrary constants.
Formulas (15.10.2.29) and (15.10.2.31) define a solution to equation (15.10.2.16) for arbitrary f (w) The function w(z) is described by equation (15.10.2.30) with constraint (15.10.2.31) To the special case a = d =0
there corresponds a traveling-wave solution and to the case b = c = d =0 , a self-similar solution.
Trang 715.10.2-3 Second-order equations of general form.
Consider a second-order equation of the general form
F(x, t, w, w x , w t , w xx , w xt , w tt) =0 (15.10.2.32) with a first-order differential constraint
G(x, t, w, w x , w t) =0 (15.10.2.33) Let us successively differentiate equations (15.10.2.32) and (15.10.2.33) with respect to both variables so as to obtain differential relations involving second and third derivatives
We get
Dx F =0, Dt F =0, Dx G =0, Dt G =0, Dx[Dx G]=0, Dx[Dt G]=0, Dt[Dt G]=0
(15.10.2.34) The compatibility condition for (15.10.2.32) and (15.10.2.33) 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 (15.10.2.32)–(15.10.2.34) In doing so, we obtain an expression of the form
H(x, t, w, w x) =0 (15.10.2.35)
If the left-hand side of (15.10.2.35) is a polynomial in w x, then the compatibility conditions result from equating the functional coefficients of the polynomial to zero
15.10.3 Second- and Higher-Order Differential Constraints
Constructing exact solutions of nonlinear partial differential equations with the help of second- and higher-order differential constraints requires finding exact solutions of these differential constraints The latter is generally rather difficult or even impossible For this reason, one employs some special differential constraints that involve derivatives with respect to only one variable In practice, one considers second-order ordinary differential
equations in, say, x and the other variable, t, is involved implicitly or is regarded as a parameter, so that integration constants depend on t.
The problem of compatibility of a second-order evolution equation
∂w
∂t =F1
x, t, w, ∂w
∂x,∂
2w
∂x2
with a similar differential constraint
∂w
∂t =F2
x, t, w, ∂w
∂x,∂
2w
∂x2
may be reduced to a problem with the first-order differential constraint considered in
Paragraph 15.10.2-1 To that end, one should first eliminate the second derivative w xxfrom the equations Then the resulting first-order equation is examined together with the original equation (or the original differential constraint)