#### A new equation and exact solutions describing focal fields in media with modular nonlinearity

A new equation and exact solutions describing focal fields in media with modular nonlinearity
Oleg V. Rudenko 0 1 2 3
Claes M. Hedberg 0 1 2 3
0 O. V. Rudenko Nizhni Novgorod State University , Nizhni Novgorod , Russia
1 O. V. Rudenko Department of Physics, Moscow State University , Moscow , Russia 119991
2 O. V. Rudenko Schmidt Institute of Physics of the Earth, Russian Academy of Sciences , Moscow , Russia
3 O. V. Rudenko Prokhorov General Physics Institute, Russian Academy of Sciences , Moscow , Russia
Brand-new equations which can be regarded as modifications of Khokhlov-ZabolotskayaKuznetsov or Ostrovsky-Vakhnenko equations are suggested. These equations are quite general in that they describe the nonlinear wave dynamics in media with modular nonlinearity. Such media exist among composites, meta-materials, inhomogeneous and multiphase systems. These new models are interesting because of two reasons: (1) the equations admit exact analytic solutions and (2) the solutions describe real physical phenomena. The equations model nonlinear focusing of wave beams. It is shown that inside the focal zone a stationary waveform exists. Steady-state profiles are constructed by the matching of functions describing the positive and negative branches of exact solutions of an equation of Klein-Gordon type. Such profiles have been observed many times during experiments and numerical studies. The non-stationary waves can contain singularities of two types: discontinuity of the wave and of its derivative. These singularities are eliminated by introducing dissipative terms into the equations-thereby increasing their order. Mathematics Subject Classification 35C07 · 35G20 · 35K55 · 74J30 · 74J40 · 76E30
Nonlinear partial differential equation; Exact solution; Modified KZ-OV; Modular nonlinearity; Bimodular media; Focusing; Highintensity focused ultrasound; HIFU
1 Introduction
Wave beams in quadratic nonlinear media are governed
by the Khokhlov–Zabolotskaya (KZ) equation,
published first in Ref. [1]:
Here p is the acoustic pressure, x is the axial
coordinate along which the wave propagates, r is the radial
polar coordinate of the beam, τ = t − x /c is the time
in the reference system moving with sound velocity c.
The nonlinear parameter of the medium is ε. The
history of construction of this equation and main physical
results following from it are reviewed in Ref. [2]. The
technique of derivation is based on a combination of
the slowly varying profile method and the quasi-optical
approximation and is described in detail in books [3,4].
The model (1) is widely used for underwater [3] and
medical [4] applications.
Modern ultrasonic technologies and medical devices
are based mainly on high-intensity focused ultrasound
(HIFU). Focusing is a natural way to concentrate
energy, but also leads to strong nonlinear phenomena.
When converging at the focal point, shock fronts form
in the wave and a large nonlinear absorption appears. At
the same time, the diffraction effects in the focal region
are important. These diversified phenomena
accompanying the nonlinear focusing in HIFU are described in
the review [5] (Sect. 5) where a detailed list of
references is given. Only those which contain results needed
for our further analysis will be discussed here.
In Ref. [6], the coefficient of concentration of energy
is calculated. It usually decreases with increasing
intensity because of nonlinear decay of the wave (see the
review by Naugol’nykh [7]). However, nonlinearity is
sometimes able to increase the concentration due to the
sharper focusing of higher harmonics [8,9]. Ostrovskii
and Sutin analyzed this phenomenon [9] on the base of
a stage-by-stage approach, which goes as follows.
Two stages can be distinguished at sharp focusing
(see Fig. 1), at two ranges of the x -coordinate. In the
first stage, which extends from the surface of
concave source to the beginning of focal area, the wave
is spherically converging. It is subjected to only
nonlinear distortion–diffraction is negligible. At the second
stage around the focal point within the short waist of
length l∗, the wave front becomes plane, and the wave
profile distorts almost exclusively due to the phase
shifts between harmonics, which appear due to the
lowfrequency dispersion caused by diffraction.
As is shown in Fig. 1, the interesting waist region has
a form which is close to cylindrical. For beams round
in cross section, the length of this cylinder is l∗, and its
base radius is a∗ [3,4]:
l∗ =
2 R2
a R
R, a∗ = l
d
In this formula ld = ωa2/2c is the diffraction length,
R is the radius of curvature of the source (or focal
distance) and a is the initial value of the radius of the beam.
Hereafter the focusing will be considered strong, and
Fig. 1 Scheme of sharp focusing inside a bimodular material
containing cracks and illustration of notations used in the text
the diffraction weak. In this case, when ld R, one
can generate the most intense fields possible.
The cylindrical volume is the focal ray tube in which
the nonlinear beam with plane fronts propagates. Based
on the ideas expressed above, the following
mathematical model was derived in Ref. [10] for a wave in a
tube:
∂ ∂ p ε ∂ p2 2c
∂τ ∂ x − c3ρ ∂τ 2 = − a∗2 p. (2)
Equation (2) is valid within the waist. One can derive
it straight from the KZ (1) by formally putting the
parabolic radial dependence of the field near the axis
(r → 0) to:
r 2
p(x , τ, r ) = 1 − a2 p(x , τ ). (3)
∗
Substituting (3) into (1) and limiting the analysis to the
paraxial region (r → 0), one can derive the 1D model
(2). This model dramatically simplifies the qualitative
analysis of nonlinear low-frequency diffraction.
Evidently, the 1D Eq. (2) is much simpler to solve than the
2D Eq. (1) both analytically and numerically.
Equation (2) obtained in Ref. [10] for the focal wave
field has a quite general meaning and is a popular model
of nonlinear dynamics. It is often called the Ostrovsky–
Vakhnenko (OV) Eq. [11] and was initially constructed
for other physical systems, like for internal waves in
rotating ocean [12], and for finding solutions to
mathematical physics models similar to solitons [13]. An
analysis of an even more general OV equation with
varying coefficients was done in Ref. [14].
This current article is associated with Ref. [10], but
here appears fundamentally different mathematics. The
new equation that is solved here describes a medium
with a modular nonlinearity, as distinct from the more
common quadratic nonlinear medium.
The form of the wave in the focal area was calculated
in Ref. [10] for a common medium. At high intensities
of the focused wave, the profile is depicted by a periodic
sequence of arcuate sections having singularities of the
derivative at the points of matching—typical for waves
in nonlinear systems with low-frequency dispersion.
Shockwaves containing steep fronts vary their shape
while passing the focal area and they cannot be
stationary. The maximum possible focal fields were estimated
to be 50 kW/cm2 [10], which is in agreement with both
experiments and computer simulations.
In this paper, strongly distorted nonlinear forms of
waves are described. These may in some aspects remind
of solitons which in conservative systems result from
competition between nonlinearity and dispersion [15,
16]. Furthermore, solitons are single pulses, while this
article treats periodic waves.
2 The modified KZ–OV equation for a bimodular
material
Studies of plane waves in media with modular
nonlinearity described by modified Burgers-type equations
started recently [17]. These media have different
elastic constants for the tensile and compressional
deformations. They are bimodular materials, and examples are
armed polymers and concretes (see [18], Ch.1, Sec.10),
as well as cracked solids, and gas–liquid and
multiphase media. The difference in elasticity at change of
sign of deformation can be 10–15% [18]. The
corresponding equations are formally linear when the wave
function conserves its sign, i.e., for purely p > 0 or
for purely p < 0, and the nonlinear phenomena are
displayed only for waves containing both compression
and rarefaction areas.
The analogue to Eq. (2) for the bimodular medium
can be obtained by the replacement of the quadratic
nonlinearity p2/2 in (2) by the module nonlinearity
| p|:
The scheme of derivation of (4) is standard [3,4]. For
example, for a fluid the equations of motion (Euler) and
continuity for weak disturbances of pressure and
density can be reduced to the following differential
equation:
The equation of state is here:
The density variation ρ is gotten rid of by combining
(5) and (6). Thereafter the model is transformed to a
coordinate system [x , τ = t − x /c], moving along x
with the sound velocity c. The wave distortion is slow
along the x axis (∼μ x ) and in the transverse
direction r (∼√μ r ). This slow variation is caused by weak
influence of nonlinear and diffraction effects and these
terms are small on the order of μ 1. Neglecting
terms of second and higher orders of the small
parameter μ, and restricting attention to the focal region, one
obtains the evolution Eq. (4). By analogy, this equation
can be derived for solid media from the equations of
the theory of elasticity.
It is convenient to reduce (4) to dimensionless form.
Use the notations:
x gω p
z = lNL = 2c x , θ = ωτ, V = p0 ,
The meanings of the constants ω and p0 are the typical
frequency and amplitude of initial signal, and lNL is
the nonlinear length. The number γ 2 is the similarity
criterium. At γ 2 1, diffraction dominates, and at
γ 2 1, nonlinearity is stronger. The notations (7)
reduce (4) to the following equation:
3 Stationary solution to the modified KZ–OV Equation (8) can be represented by a pair equivalent to Klein–Gordon’s equations:
∂∂z2∂VT±± = −γ 2V , T± = θ ± z. (9)
The upper signs are for V > 0, and the lower signs
are for V < 0. The linear equations (9) can be
easily solved separately, but minor complications appear
when linking the positive and negative branches of the
solution to Eq. (9).
The positive branch is described by one of the two
formulas:
V+ = C1+ cos δ+ T+ + δ2 z
+
γ 2
V+ = D1+ cosh δ+ T+ − δ2 z
+
γ 2
+ D2+ sinh δ+ T+ − δ2 z .
+
For the negative branch, by analogy:
V− = C1− cos δ− T− + δ2 z
−
γ 2
V− = D1− cosh δ− T− − δ2 z
−
γ 2
+ D2− sinh δ− T− − δ2 z , (11)
−
where δ, C and D are constants.
In (10) and (11), one can see that the waves V+ and
V− can propagate with equal velocities. For example,
equating the phases from the second formula in (10)
and the first formula in (11) one obtains:
Consequently, for any given number γ 2 the constants
δ2 and δ2 can be determined to fullfil relation (13). This
+ −
means that the branches V+ and V− being linked at an
initial z = z0 will be continuously sewn together during
the propagation of the wave, i.e., at all subsequent z.
To calculate the form of stationary wave, it is
convenient to seek for the steady-state solutions to Eq. (8)
in the form:
For beams having their maximum amplitude on the
axis, diffraction increases the velocity of propagation,
so it must be that β > 0 [19]. The substitution (14)
reduces (8) to the ordinary differential equation:
d2V γ 2
dθ 2± ∓ 1 ± β V± = 0. (15)
Its two solutions, which can be linked through the use
of the formulas (10)–(13), can be written as:
V− = C1− cos √1γ +T β + C2− sin √1γ +T β ,
V+ = D1+ cosh √1γ −T β + D2+ sinh √1γ −T β .
These solutions are valid for 0 < β < 1.
For the convenience to perform the matching
procedure, we will combine the solutions (16) to construct
an even function. One can do that by shifting along
T because we are solving the autonomous equation
whose coefficients have no T -dependence. The
equation is nonlinear, but is homogeneous in the variable
V , which means that all integration constants in (15)
can be multiplied by the same number. Therefore, the
solution (16) can be rewritten in a more simple form:
V− = − cos √1γ +T β , 0 < T < T1 = 2πγ 1 + β,
V+ = D sinh γ √(T1−− Tβ1) , T1 < T < π. (17)
Both branches of the solution (17) are continuously
matched in the point T1. In addition, continuity of
derivative in the same point is required and also that
the mean-over-period value of the wave field is zero:
T1 π
V− dT +
V+ dT = 0.
0 T1
The formula (18) follows directly from Eq. (8). The
physical meaning of (18) is the conservation of linear
momentum of a volume of the medium in which the
wave propagates [4].
These additional conditions make it possible to
calculate the constant D in formula (17), as well as the
connection between parameters γ and β. The final form
of this solution is:
V− = − cos √1γ +T β , 0 < T < T1 = 2πγ 1 + β,
V+ =
Fig. 2 Profiles of a periodic wave near the focus constructed by
the matching of negative trigonometric and positive hyperbolic
functions. Curves 1, 2 and 3 correspond to different velocities of
propagation at parameters β = 0, 0.8, 0.99
The solution (19) for one period of wave is shown
in Fig. 2. Curves 1, 2, 3 correspond to different values
of the parameter β = 0, 0.8, 0.99. One can see that the
duration of the positive part is shorter than the negative.
Moreover, the branch V+ has a sharp maximum, and
the derivative has a discontinuity at T = π . This
singularity would evidently be smoothed by the presence
of dissipation. The negative branch V− is less steep but
has longer duration. The maximum of V+ exceeds the
maximum of module of the negative branch |V−|:
(V+)MAX =
> |V−|MAX = 1.
This difference increases with increase in velocity of
propagation, i.e., when β → 1.
Wave profiles such as those shown in Fig. 2 have
been observed many times during experiments and
numerical studies of the KZ equation near the focus and
have been described by approximate analytical
methods as well. However, the exact analytical solution for
modular nonlinearity is found only in this present work.
The mean intensity of the wave (19) can be defined
through the integral over the period of the square of the
function:
Another periodic solution can be constructed by
matching the trigonometric functions for both positive
V+ and negative V− branches. The joining of such
functions is possible at higher values of velocities β > 1.
In this case the analogue of (17) is:
γ T π
V− = − cos √β + 1 , 0 < T < T1= 2γ 1 + β,
V+ = D sin γ √(Tβ−−T11) , T1 < T < π, (22)
This solution is continuous in the point T1. By means
of the additional condition (18), one of the constants is
calculated:
sin γ √(T −T1)
β + 1 β−1 . (23)
D = β − 1 1 − cos γ √(π−T1)
β−1
The connection between constants β and γ follows
from the requirement for the period to be 2π :
With account for the connections (23), (24), the final
form of the solution (22) is:
V− = − cos
By analogy with the main characteristics of the
solution (19), namely its maximum value (20) and its mean
intensity (21), the corresponding values for the solution
(25) are:
V 2 = 21 ν, (V+)MAX = ν. (26)
In this case too, the maximum of V+ exceeds the
maximum of the absolute value of the negative branch |V−|.
This difference increases with decrease in velocity of
propagation, i.e., when β → 1 (see Fig. 3).
4 Non-stationary solution
The case to consider now is when the wave arriving at
the input of the waist does not satisfy the conditions
of linking, periodicity and zero mean level (18). This
wave can then not be stationary, i.e., it cannot propagate
with an unchanged shape. As an example, consider an
Fig. 3 Profiles of periodic wave near the focus constructed by
matching of trigonometric functions. Curves 1, 2 and 3
correspond to different velocities of propagation at β = 1.6, 1.2, 1.01
harmonic initial wave and start by using the formulas
(10) and (11):
γ 2
V+ = C2+ sin δ+ T+ + δ2 z
+
γ 2
V− = C2− sin δ− T− + δ2 z
−
During propagation, at increasing z, the wave is
described by the solution:
V− = C sin θ − (1 − γ 2)z ,
V+ = C sin θ + (1 + γ 2)z .
The initial wave (28) is shown in Fig. 4 by the dashed
curve, and the solution (29) by the solid curve. The
diffraction is not too strong—as γ 2 < 1.
Fig. 4 Profile of one period of initially harmonic wave (dashed
line) and its distorted shape at the distance z (thick solid line)
As shown in Fig. 4, nonlinearity creates three
singularities during each period: the break of the function
V (θ ) itself (at θ = −γ 2z) and the breaks of its
derivative (at θ = π − (1 + γ 2)z and at θ = π + (1 − γ 2)z).
These singularities are marked by solid circles.
A discontinuity singularity occurs in the case when
V+ follows V−. A shock front appears inside the
intersection area and cuts two equally shaded areas (Fig. 4).
Another type of singularity, a discontinuity of the
derivative occurs if V− follows V+. In this case the
two parts move away from each other. One can see in
Fig. 4 how a nonlinear decay of the wave takes place,
and at the finite distance z = π the wave completely
disappears.
The Fourier series expansion of solution (29) is:
V =
2 Bn = n2 − 1 sin z + n2 − 1 sin(nz),
With increase in z, the fundamental harmonic (n = 1)
fades monotonically, giving its energy to the higher
harmonics, which are in turn transferred to the medium—a
nonlinear dissipation process. At first, the amplitudes
of the higher harmonics (n > 1) grow while receiving
energy from the fundamental wave, but thereafter they
decay as well—to zero at z = π .
5 Smoothing of singularities by dissipation The singularities shown in Figs. 2, 3 and 4 would be smoothed by dissipation, which of course exists to
some extent in any medium. To derive the smooth
solutions, it is necessary to generalize the differential
equations (4) and (8) by introducing a dissipative term of
higher order. Starting with Navier–Stokes’ equation of
motion, one can, instead of (5), derive the following
differential relation:
Here b is the dissipative coefficient, and ρ0 is the
equilibrium density of the medium. Using the equation
of state (6) and repeating the procedure of derivation
described in Sect. 2, we obtain the generalized Eq. (4):
It is reasonable to call this equation a modified
Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation
[3] because it differs from original KZK in the
replacement of the quadratic nonlinearity p2/2 by a modular
one | p|.
Using the dimensionless variables (7), Eq. (33) can
be reduced to a normalized equation [generalizing (8)]:
∂2V
The new parameter Γ is connected with the typical
dissipative length lDISS for a given medium through:
To smooth the discontinuity of a function (see shock
front in Fig. 4), the singularly perturbed problem must
be considered with Γ being a small parameter in the
highest (third) derivative in Eq. (34). As follows from
Fig. 4, the symmetric shock is located at θ = −γ 2z. If
a new temporal variable T = θ + γ 2z is introduced,
the shock will be immovable at T = 0. With the new
variable ξ = T /Γ , Eq. (34) takes a steady-state form:
One can see that at weak dissipation Γ 1 the shock
front structure is governed by the simplified equation:
The solution to (35) satisfying the two boundary
conditions at the infinity—V → V0, ξ → +∞ and
V → −V0, ξ → −∞—has the form:
VV+0 = 1 − exp − (1 − γ 2) θ +Γγ 2z ,
V− = −1 + exp (1 + γ 2) θ + γ 2z . (38)
V0 Γ
It describes the steep shock front which transforms to
an ideal discontinuity at Γ → 0. In accordance with
the matched asymptotic expansions method described
by Nayfeh [20], the result (38) is the main term of the
inner expansion, and corresponding term of the outer
expansion is determined by (29): V− = C sin(T − z),
V+ = C sin(T + z) which at T = 0 gives the necessary
value of the constant V0 = V0(z) = C sin z.
A similar problem can be solved to smooth the
derivative discontinuity. The final result, omitting the
calculating details, is given as:
1
V+ = 2 θ − θ1(z) Φ − 1 ,
Here Φ(x ) = √2π 0x exp(−ν2) dν is the error integral,
and θ1(z) = π − (1 + γ 2)z and θ2(z) = π + (1 − γ 2)z
are the positions of the singular points (see Figs. 4, 5).
With increase in the number Γ and the distance z, the
Fig. 5 Elimination of singularity (discontinuity of derivative),
appearing at the two points marked by solid circles. Smoothed
curves 1 and 2 correspond to Γ z = 0.25 · 10−2 and 10−2
V−
smoothing is enhanced (see curves for Γ z = 0.25·10−2
and 10−2 in Fig. 5).
Note that the analytical smoothing described above
is consistent with the results of numerical modeling
performed by Nazarov and co-workers for other equations
with modular nonlinearity [17, 21]. The construction
of smoothed solutions can be done using the standard
method of matched asymptotic expansions. A detailed
summary of these methods is given in the book by
Nayfeh [20].
6 Conclusion
This article contains two types of results. On the one
hand, new nonlinear equations are obtained which
generalize the previously studied models of KZ, OV and
KZK types. Solutions to these equations are found,
describing the nonlinear dynamics of waves in focal
areas. In media without dissipation the singularities that
form are of two types: discontinuities of the wave itself
and discontinuities of its derivative. Smoothing
procedures which eliminate these singularities are thereafter
indicated. This group of results is attributed to the field
of mathematical physics and nonlinear dynamics.
On the other hand, the studies of the physics of
nonlinear wave focusing, which was initiated in Ref.
[10], are here continued for high-intensity waves in
bimodular media. These materials exist in nature—one
example is cracked or granular materials [22].
Modular and other related unusual nonlinearities exist also in
composites and meta-materials, increasing the
applicative interest of these studies [23], and the mathematical
and physical aspects of the nonlinear dynamics in such
materials will be expanded in the nearest future.
The analytical methods used in this work are
standard ones, and the main difficulty encountered was that
singularities exist as long as the models do not have
damping. These singularities disappeared when
dissipation was introduced. As any real system of this type
always has a nonzero amount of dissipation, this also
made the results being closer to the physical reality.
The results here are new as they investigate the modular
material model which is based on real materials’
structures and the obtained results are qualitatively close to
their behavior.
The biggest advantage with the modular equation
investigated here is that exact analytical solutions were
obtained. There is therefore no need for numerical
results to make comparisons to and all solutions in this
paper are purely analytical.
Acknowledgements This work is supported by the Russian
Scientific Foundation Grant No. 14-22-00042.
Open Access This article is distributed under the terms of
the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits
unrestricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original author(s) and
the source, provide a link to the Creative Commons license, and
indicate if changes were made.
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