The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves
The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves
Oleg V. Rudenko 0 1 2 3 4
Claes M. Hedberg 0 1 2 3 4
0 O. V. Rudenko Nizhni Novgorod State University , Nizhni Novgorod , Russia
1 O. V. Rudenko Department of Physics, Moscow State University , 119991 Moscow , Russia
2 O. V. Rudenko
3 O. V. Rudenko Schmidt Institute of Physics of the Earth, Russian Academy of Sciences , Moscow , Russia
4 O. V. Rudenko Prokhorov General Physics Institute, Russian Academy of Sciences , Moscow , Russia
A modified equation of Burgers type with a quadratically cubic (QC) nonlinear term was recently pointed out as a new exactly solvable model of mathematical physics. However, its derivation, analytical solution, computer modeling, as well as its physical applications and analysis of corresponding nonlinear wave phenomena have not been published up to now. The physical meaning and generality of this QC nonlinearity are illustrated here by several examples and experimental results. The QC equation can be linearized and it describes the experimentally observed phenomena. Some of its exact solutions are given. It is shown that in a QC medium not only shocks of compression can be stable, but shocks of rarefaction as well. The formation of stationary waves with finite width of shock front resulting from the competition between nonlinearity and dissipation is traced. Singlepulse propagation is studied by computer modeling. The nonlinear evolutions of N and Swaves in a dissipative QC medium are described, and the transformation of a harmonic wave to a sawtoothshaped wave with periodically recurring trapezoidal teeth is analyzed.
Strongly nonlinear systems; Nonlinear partial differential equation; Exact analytical solutions; Quadratically cubic equation; Shock fronts; Nonlinear acoustics and turbulence; Exact linearizations

35C07 ·
1 Introduction
The Burgers equation was until recently the only known
nonlinear partial differential equation of the second
order which simultaneously has the two important
properties that: (i) it can be exactly linearized by a
simple transformation (using the Hopf–Cole
substitution), and (ii) it has a significant physical meaning.
Suggested initially as a model to describe turbulent
spectra [1], it became the basic mathematical model
of nonlinear waves in systems where the propagation
velocity does not depend on frequency [2]. The Burgers
equation adequately describes the physical
phenomena of highintensity wave propagation in dissipative
nondispersive media with quadratic nonlinearity. Its
predictive efficiency has been verified many times in
comparisons between experiments and numerical
simulations [3–5].
A second nonlinear partial differential equation
which can be linearized by a simple substitution was
indicated recently by the authors [6–8]. This is a
quadratically cubic (QC) Burgerstype equation. Its
derivation and physical applications and the analysis
of the corresponding nonlinear wave phenomena have
not been published prior to this current paper. Like the
usual Burgers equation, the QC equation also has a
significant physical meaning, but its manifestations are
completely different.
A version of Burgers’ equation is written here for a
nonlinear acoustic wave as [2]:
∂ p
∂ z = c3ρ
p
∂ p b ∂2 p
∂τ + 2c3ρ ∂τ 2
.
Here p is the disturbance of pressure, ρ is the
equilibrium density, is the nonlinear coefficient, τ =
t − z/c is the time measured in a coordinate system
accompanying the wave along the zaxis with sound
velocity c, and b is the effective dissipation which
depends on the shear and bulk viscosities and the
thermal conductivity [2].
For the mathematical analysis, it is convenient to
rewrite the Burgers equation in a dimensionless form:
∂ V ∂ V ∂2V
∂ Z = V ∂θ + Γ ∂θ 2
The following normalized variables are used in formula
(
2
):
z p
Z = zSH , θ = ωτ, V = p0 ,
Γ = zzDSIHSS = 2bωp0 .
Here ω and p0 are the typical frequency and amplitude
of the initial wave, and zSH and zDISS are typical shock
formation and dissipation lengths defined by
zSH =
c3ρ 2c3ρ
ωp0 , zDISS = bω2
The dimensionless parameter combination Γ —known
as the acoustical Reynolds number or Goldberg’s
number—is the only similarity criterion for
onedimensional nonlinear waves in dissipative media. At
Γ 1 dissipation dominates over nonlinearity, and at
Γ 1 nonlinearity is stronger.
The QC Burgerstype equation can be written, by
using the same variables as in (
3
), in the following
dimensionless form [6–8]:
∂ V 1 ∂ ∂2V
∂ Z = 2 ∂θ (V V ) + Γ ∂θ 2
.
The difference between Eqs. (
5
) and (
2
) is that a QC
nonlinearity V V is present in (
5
), instead of the
quadratic nonlinearity V 2 of the Burgers equation. The
term V V is functionally rather similar to V 3 in its
symmetry. Therefore one can say that the usual cubic
nonlinearity V 3 [9,10] is modeled here by the
piecewise quadratic relation V V . This function is
continuous, as is its first derivative, while the second derivative
has a singularity at V = 0. Equation (
5
) is possible to
linearize by using the substitution [8]
(
1
)
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
∂
V  = 2Γ ∂θ ln U ⇒
∂U ∂2U
∂ Z = Γ ∂θ 2 + CU.
Therefore Eq. (
5
) can be solved much more
easily than the Burgerstype equation with the common
cubic nonlinearity V 3. But the formal replacement
V 3 → V V  is useful not only for the qualitative
analysis of cubic nonlinear phenomena. It is more crucial to
know that real systems with the nonlinearity V V  exist.
Examples of such will be provided in the next section.
2 Examples of physical systems with quadratically
cubic nonlinearity
The first example is related to a strong shear wave
propagating in the structure shown in Fig. 1. It consists of
rigid plates located periodically along the zaxis. The
plates are only allowed to move up and down in the x
direction (Fig. 1a). At equilibrium, the center of each
plate lies on the zaxis. They are connected by linear
elastic forces (Hookean springs) as shown in Fig. 1b.
Fig. 1 a Shear wave in the
structure of planeparallel
plates interconnected by
elastic forces, b image patch
of three neighboring plates
and springs is given for the
derivation of equation of
motion
The displacement of the center of the nth plate up from
zaxis is ξn(t ). An initial displacement of one or more
of the plates from equilibrium leads to a shear wave
propagating along the zaxis. In deriving an equation
of motion, we consider Fig. 1b where a is the spatial
period of the structure, and d is the thickness of the
deformable layer where the elastic force acts. Let us
now consider three neighboring plates, each having the
same mass M . The equation of motion for the plate
with number n is:
M dd2tξ2n = Fn−1 + Fn+1.
The force acting from plate number n − 1 is:
Fn−1 = −k
(ξn − ξn−1)2 + d2 − d cos α.
Here k is the stiffness coefficient of the linear spring.
As we are interested in the projection of the restoring
force on the x axis, the factor cos α appears:
cos α =
ξn − ξn−1
(ξn − ξn−1)2 + d2
The displacements are considered to be small in
comparison with the thickness d of the elastic layer. In this
approximation the force (
8
) is:
Fn−1 = −k
(ξn −2dξ2n−1)3 .
The force acting from plate number n + 1 is calculated
analogously, and Eq. (
7
) takes the form:
M dd2tξ2n = −k
(ξn − ξn−1)3
2d2
− k
(ξn −2dξ2n+1)3 .
If all plates except the plate with number n are fixed,
this equation becomes
dd2tξ2n + Mkd2 ξn3 = 0.
It is similar to the wellknown Duffing equation but
is missing the linear term kξn. This ordinary
differential equation, which lacks a transition to linear
vibration at infinitesimally small amplitude, was used by W.
Heisenberg in his nonlinear quantum field theory [11].
We will pass now from the discrete chain of Eq. (
11
)
to the continuum limit. Let the wavelength be much
longer than the period a of the structure in Fig. 1.
Assuming that in (
11
)
ξn = ξ(z), ξn+1 = ξ(z + a), ξn−1 = ξ(z − a),
(
13
)
and expanding the displacements (
13
) in powers of a:
∂ξ 2 ∂2ξ
ξn±1 = ξ(z ± a) ≈ ξ(z) ± a ∂ z + a ∂ z2
we derive the nonlinear partial differential equation
∂2ξ
∂t 2 = 3β
∂ξ
∂ z
2 ∂2ξ
k a4
∂ z2 , β = M d2 .
It is now convenient to pass from Eq. (
15
) written for
the displacement ξ to an equation for the dimensionless
variable ζ = ∂ξ /∂ z, which describes the deformation
of the structure:
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
(
12
)
(
14
)
(
15
)
(
16
)
∂2ζ ∂2ζ 3
∂t 2 = β ∂ z2
.
The linear term is missing in Eq. (
16
). Consequently,
there is no limiting transition to a linear wave
equation even for very weak disturbances. Therefore, in
accordance with the classification suggested in Ref. [7],
Eq. (
16
) describes a strongly nonlinear wave of the third
type.
∂ζ
∂t =
A simpler equation of the first order corresponding
to the secondorder Eq. (
16
) is:
This can be proven by differentiating both sides of (
17
)
with respect to t , and then switching the t derivative
to the zderivative in the righthand side using Eq. (
17
)
once more. Thus, the existence of QC nonlinearity (
17
)
for nonlinear shear waves is shown. This type of
nonlinearity is essential for shear waves in soft biological
tissues where a quadratic nonlinearity does not exist
because of symmetry reasons, and the stress–strain
relationship has no linear region [12].
Let us now briefly discuss other physical systems
with QC nonlinearities. The nonlinear acoustic
properties of an orifice drilled in a plate were studied in
Ref. [13]. It was shown experimentally how the relation
between pressure and velocity approaches a quadratic
law at large disturbances: p ∼ u2 . Because the
velocity reverses its sign during oscillation, this relation
must be rewritten as p ∼ u u [13]. In the Cauchy–
Lagrange integral for the case of a potential
oscillating flow, an equivalent term with an absolute value
would appear. More examples of general relations for
obstacles in oscillating flows are known in engineering
hydraulics [14]. In general, the pressure disturbance is
the sum of two terms. The first term pAC is caused
by the compressibility of the fluid, and the second term
pHY D is connected with the oscillating flow around the
obstacle:
p = pAC + pHYD = c2ρ + γρuu.
(
18
)
Here γ is the coefficient of hydraulic resistance which
depends on the shape of the body placed in the flow
[14], and ρ is the density disturbance caused by the
acoustic wave. The QC nonlinearity also describes the
nonlinear loss in the throat of Helmholtz resonator
operating as a highintensity sound absorber [15], and
some models of dry friction are based on QC
nonlinearity [16].
Interesting manifestations of QC nonlinearity were
experimentally observed in solids. In grainy media the
measured amplitude of the third harmonic depends
on the squared amplitude of fundamental frequency
wave (∼ p02) and linearly on the distance traversed in
the medium (∼z). This may be compared to a
normal quadratic nonlinear medium where the third
har1/2
(
19
)
(
20
)
monic governed by Burgers equation is proportional
to the cube of the amplitude of the fundamental
harmonic (∼ p03) and grows with distance as z2. The
unusual behavior of polycrystalline aluminum alloy
was explained by nonlinear friction at the grain
boundaries [17]. A theoretical explanation of such
dependencies (∼ p02 and ∼z) is not difficult.
The series expansion of the exact solution of QC
Eq. (
5
) at Γ = 0 was calculated in Ref. [6]:
2
nπ − En(n Z )
2
+ Jn2(n Z )
V =
∞
[1 − (−1)n] n2Z
n=1
sin(nθ + φn(Z )).
The derivation of an analogous expansion for a truly
cubic system is given in Ref. [18]. In the expansion (
19
),
which contains only odd harmonics, En is the Weber
function and Jn is the Bessel function. The result (
19
)
represents an analog of the Bessel–Fubini solution for
quadratic nonlinearity [2]. At small nonlinear distances
Z it follows from (
19
) that:
4 4 ζ ω
V ≈ 3π Z sin(3θ ), p ≈ p02 z 3π c3ρ sin 3ωτ .
The second formula in (
20
) is written in physical
dimensional variables. One can see that the third
harmonic in a QC medium really is proportional to the
squared amplitude of the fundamental harmonic ∼ p02
and grows with distance as z1.
An additional advanced research area is connected
with the solidstate physics. Some solids like mica
contain crystal planes of heavy cells with weak bonds
between neighboring planes. These types of systems,
shown in Fig. 1, can be described by similar
mathematical models [19].
3 Selfsimilar solutions of the quadratically cubic
equation
The summary of systematic analysis of Lie group
symmetries for Burgers’ equation is given by Ibragimov
in Ref. [20]. The infinitesimal symmetries of this
equation form 5D Lie algebra stretched over the five linearly
independent operators. Unfortunately, the occurrence
of a module in the QC equation eliminates most of
these symmetries. Nevertheless, some of the
remaining symmetries generate exact solutions which carry
important physical meanings.
Let us at first consider the selfsimilar solution.
Using the substitution following from the dilation
symmetry group:
V =
2Γ
Z
Ψ
θ
ξ = √2Γ Z
.
we can reduce (
5
) to an ordinary differential equation:
d2Ψ dΨ dΨ
dξ 2 + 2Ψ  dξ + ξ dξ + Ψ = 0.
Integrating once, we get
dΨ
dξ + Ψ Ψ + ξ Ψ + C = 0.
By transformation of variables Ψ  =
Eq. (
23
) can be reduced to linear form:
Y /Y the
Y
+ ξ Y + C Y sgn(Ψ ) = 0.
The solution of (
24
) at C = 0 [2] describes a unipolar
pulse for which Ψ  = Ψ . Therefore, this solution
satisfies Burgers equation as well and does not deal with any
QC nonlinearity. The simplest nontrivial QCspecific
solution corresponds to C = 1, where
ξ 2
− 2
Y1 = exp
Ψ > 0,
Ψ < 0.
Y2 = C2ξ + exp
C1 +
ξ 2
− 2
0
ξ
+ ξ
exp
0
ξ
By matching the two branches of solution (
25
) we can
determine the arbitrary constants C1 and C2. Let both
branches (
25
) vanish in some point ξ0, i.e., Y1(ξ0) =
Y2(ξ0) = 0. The derivatives at ξ = ξ0 must be equal
to:
dΨ
dξ ξ0 = −C = −1,
1 d2Y1 1 d2Y2
Y1 dξ 2 = − Y2 dξ 2 = −1.
These matching conditions lead to the following
relations between the constants C1, C2, and ξ0:
(
21
)
(
22
)
(
23
)
(
24
)
dt,
(
25
)
(
26
)
.
0
is the error integral. By specifying the matching point
ξ0, we can calculate the constants C1 and C2 and
construct the analytical solution (
25
), (27). This result is
shown in Fig. 2. At small values of ξ0 (curve 1), the
matching point is close to the origin of coordinates.
The shape of the corresponding single pulse is similar
to a nonsymmetric Nwave. The positive area is
bigger than the negative one and contains a smooth shock
of compression at the leading front. With increase in
ξ0, the negative area becomes bigger, and a rarefaction
front appears, which increases in steepness. The
behavior and structure of the fronts are clarified by means of
an exact solution given below.
The next Lie group symmetry passed on from
Burgers’ equation to the QC equation is the translation
symmetry. This will generate a stationary wave conserving
its shape during propagation.
4 Stable and unstable shock waves of compression
and rarefaction in a quadratically cubic medium
Some physically interesting solutions to Eq. (
5
)
describe shock waves where the shock width is
controlled by dissipation. These solutions can be obtained
from the invariance with respect to time translation
group and can be sought for as
V (Z , θ ) = V (θ∗ = θ + α Z ).
Substitution of (28) into (
5
) transforms it from a partial
differential equation to an ordinary:
Γ
dV 1 1 2
dθ∗ + 2 V V − αV = 2 α .
The constant α = √2 − 1 ≈ 0.414 is determined here
from the boundary condition for the shock of
compression: V (θ∗ → ∞) → 1. For negative values of V
Eq. (29) takes form:
Γ
dV 1
dθ∗ = 2 (V + α)2.
The second boundary condition must be V (θ∗ →
−∞) → −α. Only these compressional shock waves
can be stable. The solution to (30) which satisfies the
condition V (θ∗ = θ0) = 0, where θ0 is an indefinite
constant, is
V = α
2 θ∗ − θ0
2Γ
−∞ < θ∗ < θ0.
1 − α θ∗ − θ0 −1 ,
2Γ
For positive V , Eq. (29) takes the form:
Γ
ddθV∗ + 21 (V − α)2 = α2.
Its solution for the boundary condition V (θ∗ → ∞) →
1 is:
V = α 1 +
√2 tanh α√2 θ∗
2Γ
,
θ0 < θ∗ < ∞.
The complete solution must be continuous at θ∗ = θ0
and by matching its two branches (31) and (33), we
determine the constant to be
2θΓ0 = − α √12 arctanh
It is interesting that the derivative is smooth at V = 0.
And at θ∗ = θ0, both the function V (θ∗) and its first
derivative are continuous.
(36)
(37)
Stable shocks of both compression and rarefaction
are shown in Fig. 3 for the values of Γ = 0.2 and
Γ = 1.0. The typical front width θ ∼ Γ increases
with dissipation and decreases with nonlinearity.
The processes of shock waves approaching their
steadystate form are shown in Fig. 4a for a
compression shock, and in Fig. 4b for a rarefaction shock. Both
Fig. 4 Formation of stable fronts of compression (a) and rarefaction (b) for Γ = 0.01. Curves 1–10 correspond to distances Z =
0.01, 8, 16, 24, 32, 40, 48, 56, 64, 72
shocks initially have the shape of a symmetric jump
between −1 and +1. One can see the forerunner
moving away from the shock front. As a result of this, the
shock waves obtain asymptotic values providing
stability. More exactly, the compression shock grows with
time from −α to 1, and the rarefaction shock decreases
from α to −1.
5 An Nwave in a quadratically cubic medium
The socalled Nwave is an asymptotically universal
form of any single pulse with zero linear momentum.
During propagation the leading section of the wave is
compressed and the tail section is stretched. An Nwave
can be formed as result of explosion, or at supersonic
flight at large distances from aircraft [3]. Both the
leading and the tail shocks of a normal Nwave are
compressional. A similar problem appears in QC media.
For example, for medical diagnostics, it is necessary
to calculate the shape of a pulsed shear wave in
biological tissue caused by the radiation force of focused
ultrasound [21].
For very weak linear dissipation (Γ → 0) the
solution to the QC Eq. (
5
) can be constructed employing
a graphic approach (see details in Ref. [2]). The result
is shown in Fig. 5. Here the initial bipolar pulse
consisting of two triangular regions is shown by the dotted
line (curve 1). Curves 2, 3, and 4 correspond to
increasing distances Z . At Z = 1 (curve 2) two steep shocks
have formed—a leading shock of compression and a
tail shock of rarefaction. Further down the nonlinear
evolution (curves 3 and 4), two sections are formed.
The positive pressure shape is trapezoidal, and the
rarefaction shape is triangular. The two areas of the
trapezium and the triangle are equal. The asymptotic form of
the Nwave at Γ → 0, Z > 1 has a simple analytical
representation:
θ1 = −
√
1 √
1 + Z , θ2 = − 2
V1(θ1) = √
V2(θ2) = −
1
1 + Z
√1 + 2Z + 1
2(1 + Z )
.
, V1(θ2) =
1 + 2Z − 1 ,
√1 + 2Z − 1
2(1 + Z )
The notations used here are shown in Fig. 6, with their
positions marked by the large dots.
In Fig. 7 the evolution of the Nwave is shown for
different values of the Goldberg number. Already the
initial bipolar pulse (curves 1) contains two steep shocks.
In Fig. 7a one can still trace the development of
trape
(38)
zoidal and triangular regions, but as distinct from Fig. 5,
the dissipation causes the shocks to have a finite width.
In Fig. 7b the dissipation is stronger, and the nonlinear
aspect manifests itself much less.
6 An Swave in a quadratically cubic medium
The Swave is another asymptotically universal form of
a single pulse with zero linear momentum. Its behavior
is opposite the Nwave in that the leading section is
stretched and the tail section is compressed.
In Fig. 8 the evolution of an initial Swave containing
one shock (curves 1) is shown. Two shock fronts form
at weak dissipation (Fig. 8a). The leading front is a
rarefaction shock and the tail front is a compression
shock. One can trace the development of trapezoidal
and triangular regions, but as opposed to Figs. 5 and 7,
Fig. 7 Nwave dynamics in
a QC nonlinear medium at
finite dissipation: Γ = 0.01
(a) and for distances: 0.01,
1, 2, 4, 8, 16 (curves 1–6). b
is constructed for Γ = 0.1
and distances 0.01, 0.2, 1, 2,
4, 8 (curves 1–6)
the rarefaction is ahead of the compression. In Fig. 8b
a strong dissipation suppresses the nonlinear process.
The analytical representation for the Swave, analogous
to (38) for the Nwave, can be derived fairly easily but
is not presented here.
7 An initially harmonic wave in a quadratically
cubic medium
A continuous periodic wave having a sinusoidal shape
at the input (Z = 0) is fundamental for experiments and
applications, because electromagnetic transducers
usually generate singlefrequency vibrations. Such waves
are in quadratic nonlinear media governed by the usual
Burgers equation, which describes the transformation
of a harmonic time signal profile to sawtooth shapes.
Each period has a triangular form and contains a
compression shock at the leading front [2–5]. A similar
evolution of one period of a continuous sinusoidal input
wave in a QC medium is shown in Fig. 9. As for the
periodic waves described by the Burgers equation, a
universal sawtoothshaped profile forms at large
distances. However, for the QC wave each of the teeth
of the saw has a trapezoidal form and contains two
shock waves—one of compression and one of
rarefaction. Two additional phenomena exist for a QC wave.
First, the curves in Fig. 9 demonstrate the shift of
profile to the left, which means that the wave propagation
velocity is higher. This velocity dependence on
intensity is known for quasiharmonic waves in dispersive
Fig. 8 Swave dynamics in
a QC nonlinear medium at
finite dissipation: Γ = 0.01
(a) and for distances 0.01,
0.2, 1, 2, 4 (curves 1–5). b
is constructed for Γ = 0.04
and the same distances
Fig. 9 Transformation of
an initially harmonic wave
(curve 1) to a periodic
trapezoidal sawtoothshaped
wave in a QC medium
taking place when the
Goldberg number is
Γ = 0.01 (a), and the
smoother wave obtained for
when Γ = 0.1 (b). The
curves 1–6 correspond to
distances
Z = 0, 1, 4, 8, 16, 32
media as a selfaction effect. It exists for odd
nonlinearities, and QC is one exotic example of this
nonlinearity type. Secondly, the nonlinear energy loss in the
shock fronts seen in Fig. 9 cannot exist in dispersive
media.
8 Conclusion
In this paper the attention was focused on the possibility
of an exact linearization of the quadratically cubic (QC)
Eq. (
5
), on its exact solution, as well as on the behavior
of shock fronts and single Nwave and Swave pulses.
All of these mathematical results for QC models have
physical applications.
The profiles in Figs. 7, 8 and 9, for the
dissipation parameter Γ 1, can be constructed also by
the matched asymptotic expansion method [22]. The
solutions shown in Fig. 3 can be used as main terms of
the internal expansion, and the solution of the
quadratically cubic equation at Γ = 0 [like formula (38)] can
serve as the main term of the external expansion. After
the profiles have been described analytically, it is then
possible to calculate the spectral content, the nonlinear
loss of energy at the shock fronts, and other physical
characteristics.
Finally, we emphasize that this work is intended
to draw attention to the quadratically cubic equation
model (
5
) in the belief that detailed studies of it will
continue as other QCmodifications of wellknown
equations [6] are valuable from both mathematical and
physical points of view.
One example is to extend the group analysis of these
equations, similar to what has been done for
integrodifferential equations [23]. Wave transformations for
integrodifferential equations have been treated by
similar methods [24,25].
Another important modification can be done for
nonlinear waves in grainy media [26]. When the shear
nonlinearity is included, an inhomogeneous Burgers
equation appears. It has quadratically cubic
properties, exhibiting interesting mathematical descriptions
of new physical phenomena. We intend on submitting
these results in the nearest future.
Open Access This article is distributed under the terms of
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(http://creativecommons.org/licenses/by/4.0/), which permits
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the source, provide a link to the Creative Commons license, and
indicate if changes were made.
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