On a Whitham-Type Equation
The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2009 |
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Інститут математики НАН України
2009
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| Цитувати: | On a Whitham-Type Equation / S. Sakovich // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 21 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859665224490024960 |
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| author | Sakovich, S. |
| author_facet | Sakovich, S. |
| citation_txt | On a Whitham-Type Equation / S. Sakovich // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 21 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly.
|
| first_indexed | 2025-11-30T10:41:07Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 101, 7 pages
On a Whitham-Type Equation
Sergei SAKOVICH
Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus
E-mail: saks@tut.by
Received September 27, 2009, in final form November 05, 2009; Published online November 08, 2009
doi:10.3842/SIGMA.2009.101
Abstract. The Hunter–Saxton equation and the Gurevich–Zybin system are considered as
two mutually non-equivalent representations of one and the same Whitham-type equation,
and all their common solutions are obtained exactly.
Key words: nonlinear PDEs; transformations; general solutions
2000 Mathematics Subject Classification: 35Q58; 35C05
1 Introduction
The following Whitham-type equation
ut = 2uux − ∂−1
x u2
x (1)
was proposed recently by Prykarpatsky and Prytula in [1] as a model equation describing the
short-wave perturbations in an abstract elastic one-dimensional medium with relaxation and
spatial memory effects.
This equation (1), containing the ill-defined term ∂−1
x u2
x, was represented in [1] by the second-
order nonlinear partial differential equation
uxt = 2uuxx + u2
x, (2)
and it was shown there that (2) is integrable in the sense of possessing a bi-Hamiltonian structure,
an infinite hierarchy of conservation laws, and a Lax-type representation. Also, two finite-
dimensional reductions of (2) were obtained in [1], and they turned out to be some integrable by
quadratures dynamical system, which could be useful for deriving wide classes of exact solutions
of (2).
A different representation of (1), the hydrodynamic-type system
ut = 2uux − v, vt = 2uvx, (3)
was proposed recently by Bogoliubov, Prykarpatsky, Gucwa, and Golenia in [2]. In the first
equation of (3), the ill-defined term ∂−1
x u2
x of (1) was replaced by the new variable v, while the
time evolution of v was determined by an additional first-order equation, the second equation
of (3). It was announced in [2] that (3) is integrable in the sense of possessing a Lax-type
representation. The system (3) was called in [2] an integrable regularization of the Whitham-
type equation (1).
In the present paper, we consider the nonlinear PDEs (2) and (3) in Sections 2 and 3,
respectively. We give the references which show that these equations (2) and (3), especially the
first one of them, were known and quite well studied in the literature prior to [1, 2]. We put
a particular emphasis on the fact that the equations (2) and (3) can be completely solved by
quadratures, and we refine the derivations of their general solutions, consistently following the
mailto:saks@tut.by
http://dx.doi.org/10.3842/SIGMA.2009.101
2 S. Sakovich
way used for the Rabelo equations in [3]. We have the following reasons to re-derive the general
solutions of (2) and (3). Firstly, the different derivations of the general solutions of (2), given in
the literature, all overlooked the evident class of x-independent solutions. Secondly, the general
solution of (3) can be expressed not only in a parametric form, known in the literature, but also
in an implicit form. And thirdly, we need to get the results in uniform notations, for Section 4,
where we make a comparison of the general solutions of (2) and (3), taking into account that
these two representations of the Whitham-type equation (1) are not equivalent to each other,
and obtain all common solutions of (2) and (3) exactly.
2 The Hunter–Saxton equation
The nonlinear equation (2) is far not novel. Up to a scale transformation of its variables,
this is the celebrated Hunter–Saxton equation [4], sometimes referred to as the Hunter–Zheng
equation [5]. The Hunter–Saxton equation has been studied in almost all respects, including its
complete solvability by quadratures [4, 6, 7, 8], relationship with the Camassa–Holm equation
and the Liouville equation [6, 7], bi-Hamiltonian formulation [5, 9], integrable finite-dimensional
reductions [5, 10], global solution properties [11, 12], and geometric interpretations [13, 14], to
mention only a few of numerous publications on this equation.
In our opinion, the most important feature of the Hunter–Saxton equation is the possi-
bility to obtain its general solution in a closed form. This equation is linearizable [8], or
C-integrable in the Calogero’s terminology, but it also belongs to a subset of C-integrable
equations whose general solutions can be expressed in a closed form. Such completely sol-
vable equations of the Liouville equation’s type differ from other C-integrable equations of the
Burgers equation’s type, and from the so-called S-integrable (completely integrable, or Lax in-
tegrable) equations of the sine-Gordon equation’s type, in many respects. For example, the
Liouville equation possesses a continuum of variational symmetries (hence, a continuum of
nontrivial conservation laws) and several Lax-type representations which all turn out to be
equivalent to conservation laws [15]. Leaving a study of such properties of the Hunter–Saxton
equation for a separate publication, here we only concentrate on its general solution. The
general solution of (2) can be obtained in a parametric form, in at least three different ways
[4, 7, 8]. The derivation we give below is similar to the original one of [4], but differs from it
by a more precise treatment of the arbitrariness of the transformation involved, in the spirit
of [3].
Making the transformation
x = x(y, t), xy 6= 0, u(x, t) = a(y, t), (4)
where the function x(y, t) is initially not fixed, and using the identities
ux =
ay
xy
, ut = at −
ayxt
xy
, uxx =
ayy
x2
y
− ayxyy
x3
y
,
uxt =
ayt
xy
− ayyxt + ayxyt
x2
y
+
ayxyyxt
x3
y
, (5)
we bring the nonlinear equation (2) into the form
ayt +
a2
y
xy
= ∂y
(
(xt + 2a)ay
xy
)
. (6)
Now we see from (6) that it is expedient to fix the function x(y, t) of the transformation (4)
by the condition xt + 2a = 0, which brings the equation into a constant-characteristic form and
On a Whitham-Type Equation 3
considerably simplifies it. Doing this, we find that the transformation (4) with
a = −1
2xt (7)
relates the second-order equation (2) with the third-order equation
xytt −
x2
yt
2xy
= 0 (8)
which follows from (6) and (7).
Through the transformation (4) with (7), the general solution of the third-order equation (8)
represents the general solution of the second-order equation (2) parametrically, with y being the
parameter. Note, however, that, according to the Cauchy–Kovalevskaya theorem [16], the gene-
ral solution of (8) must contain three arbitrary functions of one variable, whereas the general
solution of (2) must contain only two arbitrary functions of one variable. This redundant arbi-
trariness in x(y, t), caused by the invariance of (8) with respect to an arbitrary transformation
y 7→ Y (y) which has no effect on u(x, t) of (2), can be eliminated by the following normalization
of the parameter y. We rewrite (8) in the form
∂t
(
x−1/2
y xyt
)
= 0,
integrate over t, and get
x−1/2
y xyt = f(y), (9)
where f(y) is an arbitrary function. For any nonzero function f(y), we can set, without loss of
generality, f = 2 in (9) by an appropriate transformation y 7→ Y (y) which does not change the
corresponding solutions of (2), where the value 2 is chosen for convenience only. The case of
f = 0 must be considered separately. Consequently, all solutions of the second-order equation (2)
are represented parametrically by all solutions of the second-order equation (9) with f = 0 and
f = 2 through the transformation (4) with (7).
The case of f = 0 in (9) is xyt = 0, which immediately leads us through (7) and (5) to ay = 0
and ux = 0, that is, to the evident class of solutions
u = τ(t) (10)
of (2), with any function τ(t). In the case of f = 2, we integrate (9) over t and get
xy =
(
t+ φ(y)
)2
, (11)
with any function φ(y). Then, integrating (11) over y and using (7) and (4), we obtain the
following class of solutions of (2), determined parametrically:
x = yt2 + 2t
∫
φ(y) dy +
∫
φ(y)2 dy + ψ(t),
u(x, t) = −yt−
∫
φ(y) dy − 1
2
ψ′(t), (12)
where y is the parameter, φ(y) and ψ(t) are arbitrary functions, and the prime denotes the
derivative. The expressions (10) and (12) together constitute the general solution of the second-
order nonlinear partial differential equation (2).
Some words are due on the obtained general solution of (2). It follows from (12) that
ux =
−1
t+ φ(y)
. (13)
4 S. Sakovich
According to this relation (13), the condition ux 6= 0 is satisfied for any function φ(y), which
proves that the class of solutions (12) does not cover solutions of the class (10). For some
unknown reasons, only the parametric expressions (12) were called the general solution of (2)
in [4, 7, 8], whereas the solutions (10) were omitted there. Also, the relation (13) makes clear
that all solutions of (2), except for those of the class (10), inevitably possess singularities of the
type ux = ±∞, when considered on the interval −∞ < t < ∞. The transformation (4), used
for obtaining the general solution of (2), is applicable everywhere outside those singularities
ux = ±∞, because the condition xy 6= 0 is satisfied due to (11). This inevitable presence of
singularities in the solutions (12) was noticed in [4]. In the next section, we show that nontrivial
solutions of the representation (3) of the Whitham-type equation (1) not necessarily contain
blow-ups of derivatives.
3 The Gurevich–Zybin system
Proceeding to the hydrodynamic-type system (3), we note that this is the one-dimensional
reduction of the Gurevich–Zybin system [17, 18] which can be completely solved by quadra-
tures [18, 19]. For an earlier appearance of (3) in plasma physics, one can consult Section 3
of [20]. In [21], a bi-Hamiltonian structure and a zero-curvature representation were found and
studied for the system (3). Below we show how to obtain the general solution of (3) in an
implicit form, following the way used in [3].
Applying the transformation
x = x(y, t), xy 6= 0, u(x, t) = a(y, t), v(x, t) = b(y, t) (14)
to the system (3), with x(y, t) being not fixed initially, we obtain
at −
(xt + 2a)ay
xy
+ b = 0, bt −
(xt + 2a)by
xy
= 0. (15)
Then we fix the function x(y, t) in (14) and (15) by the condition xt + 2a = 0, and thus get
a = −1
2xt, b = 1
2xtt, xttt = 0,
that is,
x = α(y)t2 + β(y)t+ γ(y), a = −α(y)t− 1
2β(y), b = α(y), (16)
where α(y), β(y), γ(y) are three arbitrary functions, of which at least one is non-constant due
to xy 6= 0. The expressions (14) and (16) represent the general solution of the system (3)
parametrically, with y being the parameter. An appropriate transformation y 7→ Y (y), which
has no effect on solutions u(x, t), v(x, t) of (3), may be used to fix any one of the three arbitrary
functions in (16).
This parametric general solution of (3) can be expressed in an implicit form, as follows. When
the function α(y) is non-constant, we replace a(y, t) and b(y, t) in (16) by u(x, t) and v(x, t),
respectively, then eliminate y from the resulting expressions, and thus obtain
x+ vt2 + 2ut+ µ(v) = 0, u+ vt+ ν(v) = 0, (17)
where µ(v) and ν(v) are arbitrary functions (expressible in terms of the arbitrary functions α,
β, γ). When α(y) is constant but β(y) is not, we do the same and get
x+ ξt2 + 2ut+ ρ(u+ ξt) = 0, v = ξ, (18)
On a Whitham-Type Equation 5
where ρ is an arbitrary function of its argument, and ξ is an arbitrary constant. When α(y)
and β(y) are constant but γ(y) is not, we get
u = ηt+ ζ, v = −η, (19)
where η and ζ are arbitrary constants. These expressions (17)–(19) together constitute the
general solution of the nonlinear system (3).
Unlike all nontrivial solutions of the Hunter–Saxton equation, some solutions of the Gurevich–
Zybin system (3), of the class (17), do not contain blow-ups of derivatives. Indeed, it follows
from (17) that the expression for any derivative of u or v contains only some degree of the
expression t2 + 2tν ′(v)− µ′(v) in its denominator, for example,
ux =
−t− ν ′(v)
t2 + 2tν ′(v)− µ′(v)
, vx =
1
t2 + 2tν ′(v)− µ′(v)
,
where the prime denotes the derivative. Clearly, it is possible to choose the functions µ and ν
so that t2 + 2tν ′(v)− µ′(v) 6= 0 holds on the whole interval −∞ < t <∞.
4 Discussion
The general solutions of the Hunter–Saxton equation (2) and the Gurevich–Zybin system (3)
are quite different in their structure. The general solution of (2) is given in the parametric
form (12), except for the explicit solutions (10). The general solution of (3) is given in the
implicit form (17) and (18), except for the explicit solutions (19). From this point of view, the
Hunter–Saxton equation (2) and the Gurevich–Zybin system (3) are very similar to the exp-
Rabelo equation uxt = expu− (expu)xx and the quadratic Rabelo equation uxt = 1 + 1
2(u2)xx,
respectively [3].
The nonlinear PDEs (2) and (3) were considered in [1, 2] as two well-defined representations
of the Whitham-type equation (1) which itself contains the ill-defined term ∂−1
x u2
x. Evidently,
these two representations are not equivalent to each other. The Gurevich–Zybin system (3) can
be re-written as the second-order equation
utt − 4uuxt + 4u2uxx − 2uxut + 4uu2
x = 0 (20)
for u(x, t) with the definition v = 2uux − ut for v(x, t), and this second-order equation (20)
differs from the Hunter–Saxton equation (2). For this reason, one may wonder whether the
PDEs (2) and (3) have any common nontrivial solutions at all.
It can be found easily that the compatibility condition for the equations (2) and (20) is
utt = 4u2uxx + 2uxut. (21)
Alternatively, in the variables u and v, the compatibility condition for the PDEs (2) and (3) is
vx = u2
x. (22)
One can find all common solutions of (2) and (3) by applying the condition (21) to the ge-
neral solution (10) and (12) of the Hunter–Saxton equation, or, alternatively, by applying the
condition (22) to the general solution (17)–(19) of the Gurevich–Zybin system. Using the con-
dition (21), we get
τ ′′ = 0 (23)
6 S. Sakovich
from (10), and
ψ′′′ = 0 (24)
from (12). Using the condition (22), we get
µ′ + ν ′2 = 0 (25)
from (17), while (18) does not satisfy (22), and (19) satisfies (22) identically. It is quite obvious
that (10) with (23) is equivalent to (19), and that (12) with (24) is equivalent to (17) with (25).
Thus, summarizing the result in a nonrigourous way, we can say that the degree of arbitrari-
ness of common nontrivial solutions of the Hunter–Saxton equation (2) and the Gurevich–Zybin
system (3) is one arbitrary function of one variable.
Acknowledgement
The author is deeply grateful to Professor E.V. Ferapontov and Professor M.V. Pavlov for
pointing out the origin of the system (3), to the referees for their useful suggestions, and to the
Max-Planck-Institut für Mathematik for hospitality and support.
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On a Whitham-Type Equation 7
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http://arxiv.org/abs/nlin.SI/0412072
http://arxiv.org/abs/nlin.SI/0401009
1 Introduction
2 The Hunter-Saxton equation
3 The Gurevich-Zybin system
4 Discussion
References
|
| id | nasplib_isofts_kiev_ua-123456789-149083 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T10:41:07Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sakovich, S. 2019-02-19T17:06:32Z 2019-02-19T17:06:32Z 2009 On a Whitham-Type Equation / S. Sakovich // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 21 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35Q58; 35C05 https://nasplib.isofts.kiev.ua/handle/123456789/149083 The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly. The author is deeply grateful to Professor E.V. Ferapontov and Professor M.V. Pavlov for pointing out the origin of the system (3), to the referees for their useful suggestions, and to the Max-Planck-Institut f¨ur Mathematik for hospitality and support. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On a Whitham-Type Equation Article published earlier |
| spellingShingle | On a Whitham-Type Equation Sakovich, S. |
| title | On a Whitham-Type Equation |
| title_full | On a Whitham-Type Equation |
| title_fullStr | On a Whitham-Type Equation |
| title_full_unstemmed | On a Whitham-Type Equation |
| title_short | On a Whitham-Type Equation |
| title_sort | on a whitham-type equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149083 |
| work_keys_str_mv | AT sakovichs onawhithamtypeequation |