668 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
where λ/k plays the role of the wave propagation velocity (the sign of λ can be arbitrary,
the value λ = 0 corresponds to a stationary solution, and the value k = 0 corresponds to
a space-homogeneous solution). Traveling-wave solutions are characterized by the fact
that the profiles of these solutions at different time* instants are obtained from one another
by appropriate shifts (translations) along the x-axis. Consequently, a Cartesian coordinate
system moving with a constant speed can be introduced in which the profile of the desired
quantity is stationary. For k > 0 and λ > 0, the wave (15.3.2.1) travels along the x-axis to
the right (in the direction of increasing x).
A traveling-wave solution is found by directly substituting the representation (15.3.2.1)
into the original equation and taking into account the relations w
x
= kW

, w
t
=–λW

,etc.
(the prime denotes a derivative with respect to z).
Traveling-wave solutions occur for equations that do not explicitly involve independent
variables,
F

w,
∂w
∂x
,
∂w
∂t
,

∂
2
w
∂x
2
,
∂
2
w
∂x∂t
,
∂
2
w
∂t
2
,

= 0.(15.3.2.2)
Substituting (15.3.2.1) into (15.3.2.2), we obtain an autonomous ordinary differential equa-
tion for the function W (z):
F (W ,kW

,–λW

, k
2
W

,–kλW


, λ
2
W

, )=0,
where k and λ are arbitrary constants.
Example 1. The nonlinear heat equation
∂w
∂t
=
∂
∂x

f(w)
∂w
∂x

(15.3.2.3)
admits a traveling-wave solution. Substituting (15.3.2.1) into (15.3.2.3), we arrive at the ordinary differential
equation
k
2
[f(W )W

]

+ λW

= 0.

Integrating this equation twice yields its solution in implicit form:
k
2

f(W )dW
λW + C
1
=–z + C
2
,
where C
1
and C
2
are arbitrary constants.
Example 2. Consider the homogeneous Monge–Amp
`
ere equation

∂
2
w
∂x∂t

2
–
∂
2
w
∂x

2
∂
2
w
∂t
2
= 0.(15.3.2.4)
Inserting (15.3.2.1) into this equation, we obtain an identity. Therefore, equation (15.3.2.4) admits solutions of
the form
w = W (kx – λt),
where W (z) is an arbitrary function and k and λ are arbitrary constants.
15.3.2-2. Invariance of solutions and equations under translation transformations.
Traveling-wave solutions (15.3.2.1) are invariant under the translation transformations
x = ¯x + Cλ, t =
¯
t + Ck,(15.3.2.5)
where C is an arbitrary constant.
* We also use the term traveling-wave solution in the cases where the variable t plays the role of a spatial
coordinate.
15.3. TRAVELING-WAV E ,SELF-SIMILAR, AND OTHER SIMPLE SOLUTIONS.SIMILARITY METHOD 669
It should be observed that equations of the form (15.3.2.2) are invariant (i.e., preserve
their form) under transformation (15.3.2.5); furthermore, these equations are also invariant
under general translations in both independent variables:
x = ¯x + C
1
, t =
¯
t + C
2
,(15.3.2.6)

where C
1
and C
2
are arbitrary constants. The property of the invariance of specific equations
under translation transformations (15.3.2.5) or (15.3.2.6) is inseparably linked with the
existence of traveling-wave solutions to such equations (the former implies the latter).
Remark 1. Traveling-wave solutions, which stem from the invariance of equations under translations, are
simplest invariant solutions.
Remark 2. The condition of invariance of equations under translations is not a necessary condition for
the existence of traveling-wave solutions. It can be verified directly that the second-order equation
F

w, w
x
, w
t
, xw
x
+ tw
t
, w
xx
, w
xt
, w
tt

= 0
does not admit transformations of the form (15.3.2.5) and (15.3.2.6) but has an exact traveling-wave solution

(15.3.2.1) described by the ordinary differential equation
F (W , kW

,–λW

, zW

, k
2
W

,–kλW

, λ
2
W


= 0.
15.3.2-3. Functional equation describing traveling-wave solutions.
Let us demonstrate that traveling-wave solutions can bedefined as solutions of the functional
equation
w(x, t)=w(x + Cλ, t + Ck), (15.3.2.7)
where k and λ are some constants and C is an arbitrary constant. Equation (15.3.2.7) states
that the unknown function does not change under increasing both arguments by proportional
quantities, with C being the coefficient of proportionality.
For C = 0, equation (15.3.2.7) turns into an identity. Let us expand (15.3.2.7) into a
series in powers of C about C = 0, then divide the resulting expression by C, and proceed
to the limit as C → 0 to obtain the linear first-order partial differential equation
λ

∂w
∂x
+ k
∂w
∂t
= 0.
The general solution to this equation is constructed by the method of characteristics (see
Paragraph 13.1.1-1) and has the form (15.3.2.1), which was to be proved.
15.3.3. Self-Similar Solutions. Invariance of Equations Under
Scaling Transformations
15.3.3-1. General form of self-similar solutions. Similarity method.
By definition, a self-similar solution is a solution of the form
w(x, t)=t
α
U(ζ), ζ = xt
β
.(15.3.3.1)
The profiles of these solutions at different time instants are obtained from one another by a
similarity transformation (like scaling).
670 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Self-similar solutions exist if the scaling of the independent and dependent variables,
t = C
¯
t, x = C
k
¯x, w = C
m
¯w,whereC ≠ 0 is an arbitrary constant, (15.3.3.2)
for some k and m (|k| + |m| ≠ 0), is equivalent to the identical transformation. This means
that the original equation

F (x, t, w, w
x
, w
t
, w
xx
, w
xt
, w
tt
, )=0,(15.3.3.3)
when subjected to transformation (15.3.3.2), turns into the same equation in the new vari-
ables,
F (¯x,
¯
t, ¯w, ¯w
¯x
, ¯w
¯
t
, ¯w
¯x¯x
, ¯w
¯x
¯
t
, ¯w
¯
t
¯

t
, )=0.(15.3.3.4)
Here, the function F is the same as in the original equation (15.3.3.3); it is assumed that
equation (15.3.3.3) is independent of the parameter C.
Let us find the connection between the parameters α, β in solution (15.3.3.1) and the
parameters k, m in the scaling transformation (15.3.3.2). Suppose
w = Φ(x, t)(15.3.3.5)
is a solution of equation (15.3.3.3). Then the function
¯w = Φ(¯x,
¯
t)(15.3.3.6)
is a solution of equation (15.3.3.4).
In view of the explicit form of solution (15.3.3.1), if follows from (15.3.3.6) that
¯w =
¯
t
α
U(¯x
¯
t
β
). (15.3.3.7)
Using (15.3.3.2) to return to the new variables in (15.3.3.7), we get
w = C
m–α
t
α
U

C

–k–β
xt
β

.(15.3.3.8)
By construction, this function satisfies equation (15.3.3.3) and hence is its solution. Let
us require that solution (15.3.3.8) coincide with (15.3.3.1), so that the condition for the
uniqueness of the solution holds for any C ≠ 0. To this end, we must set
α = m, β =–k.(15.3.3.9)
In practice, the above existence criterion is checked: if a pair of k and m in (15.3.3.2)
has been found, then a self-similar solution is defined by formulas (15.3.3.1) with parame-
ters (15.3.3.9).
The method for constructing self-similar solutions on the basis of scaling transformations
(15.3.3.2) is called the similarity method. It is significant that these transformations involve
the arbitrary constant C as a parameter.
To make easier to understand, Fig. 15.1 depicts the basic stages for constructing self-
similar solutions.
15.3. TRAVELING-WAV E ,SELF-SIMILAR, AND OTHER SIMPLE SOLUTIONS.SIMILARITY METHOD 671
Here is a free parameter
and , are some numbers
C
km
Look for a self-similar solution
Substitute into the original equation
Figure 15.1. A simple scheme that is often used in practice for constructing self-similar solutions.
15.3.3-2. Examples of self-similar solutions to mathematical physics equations.
Example 1. Consider the heat equation with a nonlinear power-law source term
∂w
∂t
= a

∂
2
w
∂x
2
+ bw
n
. (15.3.3.10)
The scaling transformation (15.3.3.2) converts equation (15.3.3.10) into
C
m–1
∂ ¯w
∂
¯
t
= aC
m–2k
∂
2
¯w
∂ ¯x
2
+ bC
mn
¯w
n
.
Equating the powers of C yields the following system of linear algebraic equations for the constants k and m:
m – 1 = m – 2k = mn.
This system admits a unique solution: k =

1
2
, m =
1
1–n
. Using this solution together with relations (15.3.3.1)
and (15.3.3.9), we obtain self-similar variables in the form
w = t
1/(1–n)
U(ζ), ζ = xt
–1/2
.
Inserting these into (15.3.3.10), we arrive at the following ordinary differential equation for the function U(ζ):
aU

ζζ
+
1
2
ζU

ζ
+
1
n – 1
U + bU
n
= 0.
672 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Example 2. Consider the nonlinear equation

∂
2
w
∂t
2
= a
∂
∂x

w
n
∂w
∂x

, (15.3.3.11)
which occurs in problems of wave and gas dynamics. Inserting (15.3.3.2) into (15.3.3.11) yields
C
m–2
∂
2
¯w
∂
¯
t
2
= aC
mn+m–2k
∂
∂ ¯x


¯w
n
∂ ¯w
∂ ¯x

.
Equating the powers of C results in a single linear equation, m – 2 = mn + m – 2k. Hence, we obtain
k =
1
2
mn + 1,wherem is arbitrary. Further, using (15.3.3.1) and (15.3.3.9), we find self-similar variables:
w = t
m
U(ζ), ζ = xt
–
1
2
mn–1
(m is arbitrary).
Substituting these into (15.3.3.11), one obtains an ordinary differential equation for the function U(ζ).
Table 15.1 gives examples of self-similar solutions to some other nonlinear equations
of mathematical physics.
TABLE 15.1
Some nonlinear equations of mathematical physics that admit self-similar solutions
Equation Equation name Form of solutions Determining equation
∂w
∂t
=
∂
∂x


f(w)
∂w
∂x

Unsteady
heat equation
w = w(z), z = xt
–1/2
[f(w)w

]

+
1
2
zw

= 0
∂w
∂t
= a
∂
∂x

w
n
∂w
∂x


+bw
k
Heat equation
with source
w = t
p
u(z), z =xt
q
,
p =
1
1–k
, q =
k–n–1
2(1–k)
a(u
n
u

)

–qzu

+bu
k
–pu = 0
∂w
∂t
= a
∂

2
w
∂x
2
+bw
∂w
∂x
Burgers
equation
w = t
–1/2
u(z), z =xt
–1/2
au

+buu

+
1
2
zu

+
1
2
u = 0
∂w
∂t
= a
∂

2
w
∂x
2
+b

∂w
∂x

2
Potential Burgers
equation
w = w(z), z = xt
–1/2
aw

+b(w

)
2
+
1
2
zw

= 0
∂w
∂t
= a


∂w
∂x

k
∂
2
w
∂x
2
Filtration
equation
w = t
p
u(z), z =xt
q
,
p =–
(k+2)q+1
k
, q is any
a(u

)
k
u

= qzu

+pu
∂w

∂t
= f

∂w
∂x

∂
2
w
∂x
2
Filtration
equation
w = t
1/2
u(z), z =xt
–1/2
2f(u

)u

+zu

–u =0
∂
2
w
∂t
2
=

∂
∂x

f(w)
∂w
∂x

Wave equation
w = w(z), z = x/t
(z
2
w

)

=[f(w)w

]

∂
2
w
∂t
2
= a
∂
∂x

w
n

∂w
∂x

Wave equation
w = t
2k
u(z), z =xt
–nk–1
,
k is any
2k(2k–1)
(nk+1)
2
u+
nk–4k+2
nk+1
zu

+z
2
u

=
a
(nk+1)
2
(u
n
u


)

∂
2
w
∂x
2
+
∂
2
w
∂y
2
= aw
n
Heat equation
with source
w = x
2
1–n
u(z), z =y/x
(1 +z
2
)u

–
2(1+n)
1–n
zu


+
2(1+n)
(1–n)
2
u–au
n
= 0
∂
2
w
∂x
2
+a
∂w
∂y
∂
2
w
∂y
2
= 0
Equation of steady
transonic gas flow
w = x
–3k–2
u(z), z =x
k
y,
k is any
a

k+1
u

u

+
k
2
k+1
z
2
u

–5kzu

+3(3k +2)u = 0
∂w
∂t
= a
∂
3
w
∂x
3
+bw
∂w
∂x
Korteweg–de Vries
equation
w = t

–2/3
u(z), z =xt
–1/3
au

+buu

+
1
3
zu

+
2
3
u = 0
∂w
∂y
∂
2
w
∂x∂y
–
∂w
∂x
∂
2
w
∂y
2

= a
∂
3
w
∂y
3
Boundary-layer
equation
w = x
λ+1
u(z), z =x
λ
y,
λ is any
(2λ+1)(u

)
2
–(λ+1)uu

= au

The above method for constructing self-similar solutions is also applicable to systems
of partial differential equations. Let us illustrate this by a specificexample.
15.3. TRAVELING-WAV E ,SELF-SIMILAR, AND OTHER SIMPLE SOLUTIONS.SIMILARITY METHOD 673
Example 3. Consider the system of equations of a steady-state laminar boundary hydrodynamic boundary
layer at a flat plate (see Schlichting, 1981)
u
∂u
∂x

+ v
∂u
∂y
= a
∂
2
u
∂y
2
,
∂u
∂x
+
∂v
∂y
= 0.
(15.3.3.12)
Let us scale the independent and dependent variables in (15.3.3.12) according to
x = C ¯x, y = C
k
¯y, u = C
m
¯u, v = C
n
¯v. (15.3.3.13)
Multiplying these relations by appropriate constant factors, we have
¯u
∂ ¯u
∂ ¯x
+ C

n–m–k+1
¯v
∂ ¯u
∂ ¯y
= C
–m–2k+1
a
∂
2
¯u
∂ ¯y
2
,
∂ ¯u
∂ ¯x
+ C
n–m–k+1
∂ ¯v
∂ ¯y
= 0.
(15.3.3.14)
Let us require that the form of the equations of the transformed system (15.3.3.14) coincide with that of the
original system (15.3.3.12). This condition results in two linear algebraic equations, n – m – k + 1 = 0 and
–2k – m + 1 = 0. On solving them for m and n, we obtain
m = 1 – 2k, n =–k, (15.3.3.15)
where the exponent k can be chosen arbitrarily. To find a self-similar solution, let us use the procedure outlined
in Fig. 15.1. The following renaming should be done: x → y, t → x, w → u (for u)andx → y, t → x,
w → v, m → n (for v). This results in
u(x, y)=x
1–2k

U(ζ), v(x, y)=x
–k
V (ζ), ζ = yx
–k
, (15.3.3.16)
where k is an arbitrary constant. Inserting (15.3.3.16) into the original system (15.3.3.12), we arrive at a system
of ordinary differential equations for U = U(ζ)andV = V (ζ):
U

(1 – 2k)U – kζU

ζ

+ VU

ζ
= aU

ζζ
,
(1 – 2k)U – kζU

ζ
+ V

ζ
= 0.
15.3.3-3. More general approach based on solving a functional equation.
The algorithm for theconstruction of a self-similar solution, presented inParagraph 15.3.3-1,
relies on representing this solution in the form (15.3.3.1) explicitly. However, there is a

more general approach that allows the derivation of relation (15.3.3.1) directly from the
condition of the invariance of equation (15.3.3.3) under transformations (15.3.3.2).
Indeed, let us assume that transformations (15.3.3.2) convert equation (15.3.3.3) into
the same equation (15.3.3.4). Let (15.3.3.5) be a solution of equation (15.3.3.3). Then
(15.3.3.6) will be a solution of equation (15.3.3.4). Switching back to the original variables
(15.3.3.2) in (15.3.3.6),we obtain
w = C
m
Φ

C
–k
x, C
–1
t

.(15.3.3.17)
By construction, this function satisfies equation (15.3.3.3) and hence is its solution. Let us
require that solution (15.3.3.17) coincide with (15.3.3.5), so that the uniqueness condition
for the solution is met for any C ≠ 0. This results in the functional equation
Φ(x, t)=C
m
Φ

C
–k
x, C
–1
t


.(15.3.3.18)
For C = 1, equation (15.3.3.18) is satisfied identically. Let us expand (15.3.3.18) in a
power series in C about C = 1, then divide the resulting expression by (C – 1), and proceed
to the limit as C → 1. This results in a linear first-order partial differential equation for Φ:
kx
∂Φ
∂x
+ t
∂Φ
∂t
– mΦ = 0.(15.3.3.19)
674 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
The associated characteristic system of ordinary differential equations (see Paragraph
13.1.1-1) has the form
dx
kx
=
dt
t
=
dΦ
mΦ
.
Its first integrals are
xt
–k
= A
1
, t
–m

Φ = A
2
,
where A
1
and A
2
are arbitrary constants. The general solution of the partial differential
equation (15.3.3.19) is sought in the form A
2
= U(A
1
), where U(A) is an arbitrary function
(see Paragraph 13.1.1-1). As a result, one obtains a solution of the functional equation
(15.3.3.18) in the form
Φ(x, t)=t
m
U(ζ), ζ = xt
–k
.(15.3.3.20)
Substituting (15.3.3.20) into (15.3.3.5) yields the self-similar solution (15.3.3.1) with
parameters (15.3.3.9).
15.3.3-4. Some remarks.
Remark 1. Self-similar solutions (15.3.3.1) with α = 0 arise in problems with simple initial and boundary
conditions of the form
w = w
1
at t = 0 (x > 0), w = w
2
at x = 0 (t > 0),

where w
1
and w
2
are some constants.
Remark 2. Self-similar solutions, which stem from the invariance of equations under scaling transforma-
tions, are considered among the simplest invariant solutions.
The condition for the existence of a transformation (15.3.3.2) preserving the form of the given equation
is sufficient for the existence of a self-similar solution. However, this condition is not necessary: there are
equations that do not admit transformations of the form (15.3.3.2) but have self-similar solutions.
For example, the equation
a
∂
2
w
∂x
2
+ b
∂
2
w
∂t
2
=(bx
2
+ at
2
)f(w)
has a self-similar solution
w = w(z), z = xt =⇒ w


– f(w)=0,
but does not admit transformations of the form (15.3.3.2). In this equation, a and b can be arbitrary functions
of the arguments x, t, w, w
x
, w
t
, w
xx
,
Remark 3. Traveling-wave solutions are closely related to self-similar solutions. Indeed, setting
t =lnτ, x =lny
in (15.3.2.1), we obtain a self-similar representation of a traveling wave:
w = W

k ln(yτ
–λ/k
)

= U(yτ
–λ/k
),
where U(z)=W (k ln z).
15.3.4. Equations Invariant Under Combinations of Translation and
Scaling Transformations, and Their Solutions
15.3.4-1. Exponential self-similar (limiting self-similar) solutions.
Exponential self-similar solutions are solutions of the form
w(x, t)=e
αt
V (ξ), ξ = xe

βt
.(15.3.4.1)