The Euler transformation 15.2.3.9 is employed in finding solutions and linearization of certain nonlinear partial differential equations.. B¨acklund transformations may preserve the form
Trang 1With (15.2.3.5)–(15.2.3.6), we find the second derivatives
∂2w
∂x2 = J
∂2u
∂η2,
∂2w
∂x∂y = –J ∂2u
∂ξ∂η, ∂2w
∂y2 = J
∂2u
∂ξ2,
where
J = ∂
2w
∂x2
∂2w
∂y2 –
∂2w
∂x∂y
2
, 1
J = ∂
2u
∂ξ2
∂2u
∂η2 –
∂2u
∂ξ∂η
2
The Legendre transformation (15.2.3.5), with J ≠ 0, allows us to rewrite a general second-order equation with two independent variables
F
x , y, w, ∂w
∂x, ∂w
∂y, ∂
2w
∂x2,
∂2w
∂x∂y, ∂
2w
∂y2
=0 (15.2.3.7)
in the form
F
∂u
∂ξ, ∂u
∂η , ξ ∂u
∂ξ + η ∂u
∂η – u, ξ, η, J ∂
2u
∂η2, –J
∂2u
∂ξ∂η , J ∂
2u
∂ξ2
=0 (15.2.3.8)
Sometimes equation (15.2.3.8) may be simpler than (15.2.3.7)
Let u = u(ξ, η) be a solution of equation (15.2.3.8) Then the formulas (15.2.3.5) define
the corresponding solution of equation (15.2.3.7) in parametric form
Remark. The Legendre transformation may result in the loss of solutions for which J =0
Example 1 The equation of steady-state transonic gas flow
a ∂w
∂x
∂2w
∂x2 + ∂
2w
∂y2 = 0
is reduced by the Legendre transformation (15.2.3.5) to the linear equation with variable coefficients
aξ ∂
2u
∂η2 +∂
2u
∂ξ2 = 0
Example 2 The Legendre transformation (15.2.3.5) reduces the nonlinear equation
f
∂w
∂x,∂w
∂y
∂2w
∂x2 + g
∂w
∂x, ∂w
∂y
∂2w
∂x∂y + h
∂w
∂x, ∂w
∂y
∂2w
∂y2 = 0
to the following linear equation with variable coefficients:
f (ξ, η) ∂
2u
∂η2 – g(ξ, η) ∂
2u
∂ξ∂η + h(ξ, η) ∂
2u
∂ξ2 = 0
15.2.3-3 Euler transformation
The Euler transformation belongs to the class of contact transformations and is defined by
the relations
x= ∂u
∂ξ, y = η, w = xξ – u. (15.2.3.9)
Differentiating the last relation in (15.2.3.9) with respect to x and y and taking into account
the other two relations, we find that
∂w
∂x = ξ, ∂w
∂y = –∂u
Trang 2Differentiating these expressions in x and y, we find the second derivatives:
w xx = 1
u ξξ, w xy = –
u ξη
u ξξ, w yy =
u2
ξη – u ξξ u ηη
The subscripts indicate the corresponding partial derivatives
The Euler transformation (15.2.3.9) is employed in finding solutions and linearization
of certain nonlinear partial differential equations
The Euler transformation (15.2.3.9) allows us to reduce a general second-order equation with two independent variables
F
x , y, w, ∂w
∂x, ∂w
∂y, ∂
2w
∂x2,
∂2w
∂x∂y, ∂
2w
∂y2
to the equation
F
u ξ , η, ξu ξ – u, ξ, –u η, 1
u ξξ, –
u ξη
u ξξ,
u2
ξη – u ξξ u ηη
u ξξ
=0 (15.2.3.13)
In some cases, equation (15.2.3.13) may become simpler than equation (15.2.3.12)
Let u = u(ξ, η) be a solution of equation (15.2.3.13) Then formulas (15.2.3.9) define
the corresponding solution of equation (15.2.3.12) in parametric form
Remark. The Euler transformation may result in the loss of solutions for which w xx= 0
Example 3 The nonlinear equation
∂w
∂y
∂2w
∂x2 + a =0
is reduced by the Euler transformation (15.2.3.9)–(15.2.3.11) to the linear heat equation
∂u
∂η = a ∂
2u
∂ξ2.
Example 4 The nonlinear equation
∂2w
∂x∂y = a ∂w
∂y
∂2w
∂x2 (15 2 3 14 ) can be linearized with the help of the Euler transformation (15.2.3.9)–(15.2.3.11) to obtain
∂2u
∂ξ∂η = a ∂u
∂η Integrating this equation yields the general solution
u = f (ξ) + g(η)e aξ, (15 2 3 15 )
where f (ξ) and g(η) are arbitrary functions.
Using (15.2.3.9) and (15.2.3.15), we obtain the general solution of the original equation (15.2.3.14) in parametric form:
w = xξ – f (ξ) – g(y)e aξ,
x = f ξ (ξ) + ag(y)e aξ Remark. In the degenerate case a =0, the solution w = ϕ(y)x + ψ(y) is lost, where ϕ(y) and ψ(y) are
arbitrary functions; see also the previous remark.
Trang 315.2.3-4 Legendre transformation with many variables.
For a function of many variables w = w(x1, , x n), the Legendre transformation and its inverse are defined as
Legendre transformation Inverse Legendre transformation
x1= X1, ., x k–1 = X k–1, X1= x1, ., X k–1= x k–1,
x k= ∂X ∂W
k, ., x n=
∂W
∂X n, X k=
∂w
∂x k, ., X n=
∂w
∂x n,
w(x) =
n
i=k
X i ∂X ∂W
i – W (X), W(X) =
n
i=k
x i ∂x ∂w
i – w(x),
where x ={x1, , x n}, X ={X1, , X n}, and the derivatives are related by
∂w
∂x1 = –
∂W
∂X1, .,
∂w
∂x k–1 = –
∂W
∂X k–1.
15.2.4 B ¨acklund Transformations Differential Substitutions
15.2.4-1 B¨acklund transformations for second-order equations
Let w = w(x, y) be a solution of the equation
F1
x , y, w, ∂w
∂x,∂w
∂y,∂2w
∂x2,
∂2w
∂x∂y,∂2w
∂y2
=0, (15.2.4.1)
and let u = u(x, y) be a solution of another equation
F2
x , y, u, ∂u
∂x, ∂u
∂y, ∂
2u
∂x2,
∂2u
∂x∂y, ∂
2u
∂y2
=0 (15.2.4.2)
Equations (15.2.4.1) and (15.2.4.2) are said to be related by the B¨acklund transformation
Φ1
x , y, w, ∂w
∂x, ∂w
∂y , u, ∂u
∂x,∂u
∂y
=0,
Φ2
x , y, w, ∂w
∂x, ∂w
∂y , u, ∂u
∂x,∂u
∂y
=0
(15.2.4.3)
if the compatibility of the pair (15.2.4.1), (15.2.4.3) implies equation (15.2.4.2), and the compatibility of the pair (15.2.4.2), (15.2.4.3) implies (15.2.4.1) If, for some specific
solution u = u(x, y) of equation (15.2.4.2), one succeeds in solving equations (15.2.4.3) for
w = w(x, y), then this function w = w(x, y) will be a solution of equation (15.2.4.1).
B¨acklund transformations may preserve the form of equations* (such transformations are used for obtaining new solutions) or establish relations between solutions of different equations (such transformations are used for obtaining solutions of one equation from solutions of another equation)
* In such cases, these are referred to as auto-B¨acklund transformations.
Trang 4Example 1 The Burgers equation
∂w
∂t = w ∂w
∂x +∂
2w
∂x2 (15 2 4 4 )
is related to the linear heat equation
∂u
∂t = ∂
2u
∂x2 (15 2 4 5 )
by the B¨acklund transformation
∂u
∂x– 1
2uw=0,
∂u
∂t – 1 2
∂ (uw)
∂x = 0
(15 2 4 6 )
Eliminating w from (15.2.4.6), we obtain equation (15.2.4.5).
Conversely, let u(x, t) be a nonzero solution of the heat equation (15.2.4.5) Dividing (15.2.4.5) by u, differentiating the resulting equation with respect to x, and taking into account that (u t /u)x = (u x /u)t, we
u x u
t
=
u xx u
x
From the first equation in (15.2.4.6) we have
u x
u = w
u xx
u –
u x u
2
= w x
u xx
u = w x
1
Replacing the expressions in parentheses in (15.2.4.7) with the right-hand sides of the first and the last relation (15.2.4.8), we obtain the Burgers equation (15.2.4.4).
Example 2 Let us demonstrate that Liouville’s equation
∂2w
∂x∂y = e λw (15 2 4 9 )
is connected with the linear wave equation
∂2u
∂x∂y = 0 (15 2 4 10 )
by the B¨acklund transformation
∂u
∂x –∂w
∂x = 2k
λ exp *1
2λ (w + u)
+
,
∂u
∂y +∂w
∂y = – 1
kexp *1
2λ (w – u)
+
,
(15 2 4 11 )
where k≠ 0 is an arbitrary constant.
Indeed, let us differentiate the first relation of (15.2.4.11) with respect to y and the second equation with respect to x Then, taking into account that u yx = u xy and w yx = w xyand eliminating the combinations of the first derivatives using (15.2.4.11), we obtain
∂2u
∂x∂y – ∂
2w
∂x∂y = k exp*1
2λ (w + u)
+∂u
∂y +∂w
∂y
= – exp(λw),
∂2u
∂x∂y + ∂
2w
∂x∂y = 2λ k exp
*1
2λ (w – u)
+∂u
∂x –∂w
∂x
= exp(λw).
(15 2 4 12 )
Adding relations (15.2.4.12) together, we get the linear equation (15.2.4.10) Subtracting the latter equation from the former gives the nonlinear equation (15.2.4.9).
Example 3 The nonlinear heat equation with a exponential source
w xx + w yy = ae βw
is connected with the Laplace equation
u xx + u yy= 0
Trang 5by the B¨acklund transformation
u x+12βw y= 12aβ 1/2
exp 12βw
sin u,
u y– 12βw x= 12aβ 1/2
exp 12βw
cos u.
This fact can be proved in a similar way as in Example 2.
Remark 1 It is significant that unlike the contact transformations, the B¨acklund transformations are determined by the specific equations (a B¨acklund transformation that relates given equations does not always exist).
Remark 2. For two nth-order evolution equations of the forms
∂w
∂t = F1
x , w, ∂w
∂x , , ∂ w
∂x n
,
∂u
∂t = F2
x , u, ∂u
∂x , , ∂ u
∂x n
,
a B¨acklund transformation is sometimes sought in the form
Φ
x , w, ∂w
∂x , , ∂
m w
∂x m , u, ∂u
∂x , , ∂
k u
∂x k
= 0
containing derivatives in only one variable x (the second variable, t, is present implicitly through the functions
w , u) This transformation can be regarded as an ordinary differential equation in one of the dependent variables.
15.2.4-2 Nonlocal transformations based on conservation laws
Consider a differential equation written as a conservation law,
∂
∂x
F
w,∂w
∂x,∂w
∂y ,
+ ∂
∂y
G
w, ∂w
∂x, ∂w
∂y ,
=0 (15.2.4.13) The transformation
dz = F (w, w x , w y , ) dy – G(w, w x , w y , ) dx, dη = dy (15.2.4.14)
dz = ∂z
∂x dx+ ∂z
∂y dy =⇒ ∂z
∂x = –G, ∂z
∂y = F
determines the passage from the variables x and y to the new independent variables z and
ηaccording to the rule
∂
∂x = –G ∂
∂z, ∂
∂y = ∂
∂η + F ∂
∂z
Here, F and G are the same as in (15.2.4.13) The transformation (15.2.4.14) preserves the
order of the equation under consideration
Remark Often one may encounter transformations (15.2.4.14) that are supplemented with a
transforma-tion of the unknown functransforma-tion in the form u = ϕ(w).
Example 4 Consider the nonlinear heat equation
∂w
∂t = ∂
∂x
f (w) ∂w
∂x
which represents a special case of equation (15.2.4.13) for y = t, F = f (w)w x , and G = –w.
Trang 6In this case, transformation (15.2.4.14) has the form
dz = w dx + [f (w)w x ] dt, dη = dt (15 2 4 16 )
and determines a transformation from the variables x and y to the new independent variables z and η according
to the rule
∂
∂x = w ∂
∂z, ∂
∂t = ∂
∂η + [f (w)w x] ∂
∂z Applying transformation (15.2.4.16) to equation (15.2.4.15), we obtain
∂w
∂η = w2 ∂
∂z
*
f (w) ∂w
∂z
+
The substitution w =1/ureduces (15.2.4.17) to an equation of the form (15.2.4.15),
∂u
∂η = ∂
∂z
1
u2f
1
u
∂u
∂z
+
.
In the special case of f (w) = aw–2, the nonlinear equation (15.2.4.15) is reduced to the linear equation
u η = au zzby the transformation (15.2.4.16).
15.2.5 Differential Substitutions
In mathematical physics, apart from the B¨acklund transformations, one sometimes resorts to
the so-called differential substitutions For second-order differential equations, differential
substitutions have the form
w=Ψ
x , y, u, ∂u
∂x,∂u
∂y
A differential substitution increases the order of an equation (if it is inserted into an
equation for w) and allows us to obtain solutions of one equation from those of another.
The relationship between the solutions of the two equations is generally not invertible and
is, in a sense, unilateral A differential substitution may decrease the order of an equation
(if it is inserted into an equation for u) A differential substitution may be obtained as
a consequence of a B¨acklund transformation (although this is not always the case) A
differential substitution may decrease the order of an equation (when the equation for u is
regarded as the original one)
In general, differential substitutions are defined by formulas (15.2.3.1), where the
func-tion X, Y , and W can be defined arbitrarily.
Example 1 Consider once again the Burgers equation (15.2.4.4) The first relation in (15.2.4.6) can be
rewritten as the differential substitution (the Hopf–Cole transformation)
w= 2u x
u (15 2 5 1 ) Substituting (15.2.5.1) into (15.2.4.4), we obtain the equation
2u tx
u – 2u t u x
u2 = 2u xxx
u – 2u x u xx
u2 , which can be converted to
∂
∂x
1
u
∂u
∂t – ∂
2u
∂x2
Thus, using formula (15.2.5.1), one can transform each solution of the linear heat equation (15.2.4.5) into
a solution of the Burgers equation (15.2.4.4) The converse is not generally true Indeed, it follows from (15.2.5.2) that a solution of equation (15.2.4.4) generates a solution of the more general equation
∂u
∂t – ∂
2u
∂x2 = f (t)u, where f (t) is a function of t.
Trang 7Example 2 The equation of a steady-state laminar hydrodynamic boundary layer at a flat plate has the
form (see Schlichting, 1981)
∂w
∂y
∂2w
∂x∂y –∂w
∂x
∂2w
∂y2 = a ∂
3w
∂y3, (15 2 5 3 )
where w is the stream function, x and y are the coordinates along and across the flow, and a is the kinematic
viscosity of the fluid.
The von Mises transformation (a differential substitution)
ξ = x, η = w, u (ξ, η) = ∂w
∂y, where w = w(x, y), (15 2 5 4 ) decreases the order of equation (15.2.5.3) and brings it to the simpler nonlinear heat equation
∂u
∂ξ = a ∂
∂η
u ∂u
∂η
When deriving equation (15.2.5.5), the following formulas for the computation of the derivatives have been used:
∂
∂y = u ∂
∂η, ∂
∂x = ∂
∂ξ + ∂w
∂x
∂
∂η, ∂w
∂y = u, ∂
2w
∂y2 = u ∂u
∂η,
∂w
∂y
∂2w
∂x∂y – ∂w
∂x
∂2w
∂y2 = u ∂u
∂ξ, ∂
3w
∂y3 = u ∂
∂η
u ∂u
∂η
.
15.3 Traveling-Wave Solutions, Self-Similar Solutions,
and Some Other Simple Solutions Similarity
Method
15.3.1 Preliminary Remarks
There are a number of methods for the construction of exact solutions to equations of mathematical physics that are based on the reduction of the original equations to equations
in fewer dependent and/or independent variables The main idea is to find such variables and, by passing to them, to obtain simpler equations In particular, in this way, finding exact solutions of some partial differential equations in two independent variables may be reduced to finding solutions of appropriate ordinary differential equations (or systems of ordinary differential equations) Naturally, the ordinary differential equations thus obtained
do not give all solutions of the original partial differential equation, but provide only a class
of solutions with some specific properties
The simplest classes of exact solutions described by ordinary differential equations involve traveling-wave solutions and self-similar solutions The existence of such solutions
is usually due to the invariance of the equations in question under translations and scaling transformations
Traveling-wave solutions and self-similar solutions often occur in various applications Below we consider some characteristic features of such solutions
It is assumed that the unknown w depends on two variables, x and t, where t plays the role of time and x is a spatial coordinate.
15.3.2 Traveling-Wave Solutions Invariance of Equations Under
Translations
15.3.2-1 General form of traveling-wave solutions
Traveling-wave solutions, by definition, are of the form
w (x, t) = W (z), z = kx – λt, (15.3.2.1)