Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion
The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation.
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| author | Yin, J. L. Інь, Дж Л. |
| author_facet | Yin, J. L. Інь, Дж Л. |
| author_sort | Yin, J. L. |
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| description | The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered.
The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation. |
| first_indexed | 2026-03-24T02:30:17Z |
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© J. L. YIN, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8 1071
UDC 517.9
J. L. Yin (Jiangsu Univ., China)
STABILITY OF SMOOTH SOLITARY WAVES
FOR THE GENERALIZED KORTEWEG – DE VRIES
EQUATION WITH COMBINED DISPERSION*
СТІЙКІСТЬ ГЛАДКИХ ВІДОКРЕМЛЕНИХ ХВИЛЬ ДЛЯ
УЗАГАЛЬНЕНОГО РІВНЯННЯ КОРТЕВЕГА –ДЕ ФРІЗА
З КОМБІНОВАНОЮ ДИСПЕРСІЄЮ
The orbital stability problem of the smooth solitary waves in the generalized Korteweg – de Vries equation
with combined dispersion is considered. The results show that the smooth solitary waves are stable for an
arbitrary speed of wave propagation.
Розглянуто задачу про орбітальну стійкість гладких відокремлених хвиль для узагальненого рівняння
Кортевега – де Фріза з комбінованою дисперсією. Отримані результати показують, що гладкі відок-
ремлені хвилі є стійкими при довільній швидкості поширення хвиль.
1. Introduction. In order to understand the effect of nonlinear dispersion on pattern
formation as well as the formation of nonlinear structures like liquid drops etc, Rosenau
and Hyman [1] gave and studied the nonlinear dispersive Korteweg – de Vries (KdV)
equation
ut + (u2 )x + (u2 )xxx = 0. (1.1)
The nonlinear dispersion term leads to a singular solitary solution, called compacton
(i.e., solitary waves with compact support). There are many researches on compactons
(see [2 – 5]). However, the nonlinear dispersive KdV equation was not equivalent to a
Hamiltonian dynamical system. Hence this equation does not exhibit the usual energy
conservation law, Cooper, Shepard and Sodano considered instead a related generalized
KdV equation [6]
ut + uux + !(2uuxxx + 4uxuxx ) = 0 , (1.2)
which can be derived from a Lagrangian. Eq. (1.2) possesses the same terms as in
Eq. (1.1), except for the relative weights of the terms. Eq. (1.2) also admits compacton
solutions. The stability of the compacton solutions to Eq. (1.2) was considered in [7].
From the above fact, nonlinear dispersion plays a very important role in the forma-
tion of solutions. Many well-known equations present interesting singular solutions be-
cause of the nonlinear dispersion. For example, the Camassa – Holm equation [8]
ut ! uxxt + 3uux = uuxxx + 2uxuxx (1.3)
has singular solitary waves called peakons. Peakons have attracted many attentions be-
* This work is supported by the TianYuan Special Funds of the National Natural Science Foundation of
China (No. 11026169).
1072 J. L. YIN
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
cause they have a discontinuous first derivative at the wave peak. The peakons for
Eq. (1.3) are orbitally stable under small perturbations [9]. These novel solitary waves
have been researched by many authors [10 – 14].
We also know that, having linear dispersion, the well-known KdV equation and the
Boussinesq equation both have smooth solitary wave. This means linear dispersion
might be responsible for smooth solitary wave.
What is more interesting, due to the missing of the linear dispersion, all the above
three equations (1.1) – (1.3) do not have smooth solitary wave decaying to zero. If one
introduces a linear dispersion uxxx to the Camassa – Holm equation, solitary waves
appear [9].
Our first interesting is that whether the smooth solitary wave would exist if a linear
dispersion is added to (1.2). After introducing a linear dispersion term, Eq. (1.2) is mo-
dified to the generalized KdV equation with combined dispersion
ut + uux + !(2uuxxx + 4uxuxx ) + "uxxx = 0 , (1.4)
which can model the role of nonlinear dispersion and linear dispersion on pattern for-
mation as well as the formation of nonlinear structures like liquid drops. In fact, when
! = 0 , Eq. (1.4) becomes the KdV equation. When ! = 0 , it becomes Eq. (1.3).
Eq. (1.4) has the following two important conservative laws
E(u) = 1
2
u2dx
R
! , F(u) = ! 1
2
u3
3
! 2"uux2 ! #ux2
$
%&
'
()
dx
R
* . (1.5)
Another aspect absorbing our attention is that whether it is stable if the smooth soli-
tary wave exists. The stability problem of the generalized KdV equation with combined
dispersion is very attractive because of the following two points:
i) Similar to the method in [1], the generalized KdV equation with combined disper-
sion is not integrable. This suggests that the appearance of the smooth solitary waves is
probably not due to the integrability. The mechanism responsible for the coherence and
robustness of the solitary waves is still unknown. One can turn to the stability analysis
of the smooth solitary waves for help.
ii) As we know, the nonlinear term in (1.4) might lead to wave collapse. The phe-
nomena may be changed because of the existence of linear dispersion. What role does
the combination of nonlinear and linear dispersion play in the stability of the solitary
waves of this type equation? Will the solitary waves be stable?
The remainder of the paper is organized as follows. In Section 2 the existence of
smooth solitary wave solutions to Eq. (1.4) is considered. In Section 3 the orbital stabil-
ity problem of the smooth solitary waves is studied by extending the method in [15].
The result shows that the smooth solitary waves are stable for arbitrary wave speed of
propagation. The last section is the conclusions.
2. Existence of smooth solitary waves. We assume a solitary wave with speed c is
a solution to (1.4) with
u(x, t) = !c (x " ct) , (2.1)
where !c is a one variable function vanishing at infinity and c > 0 . Substituting
STABILITY OF SMOOTH SOLITARY WAVES… 1073
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
(2.1) into (1.4), we obtain
!c"cx + "c"cx + #(2"c"cxxx + 4"cx"cxx ) + $"cxxx = 0 . (2.2)
In view of the decay of !c at infinity, by integration, we obtain from (2.2)
!c"c +
1
2
"c2 + #"cxx + $(2"c"cxx + "cx2 ) = 0 . (2.3)
Multiplying by !cx both sides of (2.3) and integrating again, we have
!cx2 =
!c2 c " !c
3
#
$%
&
'(
) + 2*!c
. (2.4)
In order to study the existence of smooth solitary waves in Eq. (2.4), we will explore the
qualitative behavior of solutions of !x2 = F(!) =
!2 c " !
3
#
$%
&
'(
) + 2*!
near points where F
has a zero or a pole as follows. Firstly, if ! = 3c is a simple zero of F(!) , that is
F(3c) = 0 , !F (3c) " 0 . It is easy to obtain that !x2 = (! " 3c) #F (3c) + O (! " 3c)2( )
as !" 3c . Hence !(x) = 3c + 1
4
(x ! x0 )2 "F (3c) + O (x ! x0 )4( ) as x! x0 ,
where !(x0 ) = 3c . Secondly, if ! = 0 is a double zero of F(!) , that is F(0) = 0 ,
!F (0) = 0 , !!F (0) " 0 . We have !x2 = !2 ""F (0) +O(!3) as !" 0 . Hence
!(x) ~ " exp # x $$F (0)( ) as x! " for some constant ! .
Remark. If !(x) = " #
2$
for certain x , the solution to (2.4) is unsmooth. In or-
der to get smooth solutions, the domain of ! can not include ! "
2#
. Another condi-
tion to the existence of the solution to (2.4) is F(!) =
!2 c " !
3
#
$%
&
'(
) + 2*!
+ 0 .
Now we will show the existence of the smooth solitary wave solutions to Eq. (2.4).
According to the above analysis, there are only two cases leading to smooth solitary
wave solutions.
Case 1: ! > 0 and ! "
2#
< 0 .
In this case, we can observe that F(!) has a double zero ! = 0 , a simple zero
! = 3c , and F(!) > 0 for ! "
2#
< 0 < $ < 3c . Let ! be a solution in this interval.
We have !x " 0 as !" 0 and as !" 3c , hence ! is strictly monotonic in any
interval where F(!) > 0 . In view of (2.4), It is easy to obtain that ! is symmetric
with respect to x0 , where !(x0 ) = 3c . Note that the domain of the solution never
1074 J. L. YIN
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
touch the pole point ! "
2#
. Hence the interval of the solution to (2.4) must be the
whole real line. Also from the above analysis, !" 0 as x ! " at the double zeros
of F(!) . Therefore, when ! > 0 , and ! "
2#
< 0 , that is, ! > 0 , and ! > 0 ,
Eq. (2.4) has smooth solitary wave solutions with 0 = minx!R "(x) , 3c =
= maxx!R "(x) (see Fig. 2.1).
Fig. 2.1. The solitary wave of Eq. (1.4).
Case 2: ! < 0 and ! "
2#
> 3c .
In this case, we can observe that F(!) has a double zero ! = 0 , a simple zero
! = 3c , and F(!) > 0 for 0 < ! < 3c < " #
2$
. Similar to the discussion in Case 1,
Eq. (1.4) has the smooth solitary wave solutions with 0 = minx!R "(x) ,
3c = maxx!R "(x) when ! < 0 , ! "
2#
> 3c , namely when ! < 0 , c < ! "
6#
.
3. Stability of smooth solitary waves. To study the stability problem of the
smooth solitary wave solution to Eq. (1.4), we need the following results.
In terms of the functions E and F in (1.5), one can easily have
!F ("c ) + c !E ("c ) = 0 ,
where !E and !F are the Frechet derivatives of E and F , respectively in
H1(R) .
The linearized operator Hc of !F ("c ) + c !E ("c ) around !c is defined by
Hc = !!F ("c ) + c !!E ("c ) = # $x (% + 2&"c )$x( ) # "c # 2&"cxx + c .
We know that, since !c , !cx , !cxx " 0 exponentially fast as x ! " , smooth
solitary waves exist when the following condition (a) and (b) hold:
(a) 0 < !c < 3c ,
(b) ! < 0 and 2!"c + # > 0 (or ! > 0 and ! > 0 ).
STABILITY OF SMOOTH SOLITARY WAVES… 1075
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
It follows that the spectral equation HCv = !v can be transformed by the Liouville
transformation
!(y) = (" + 2#$c )1/4 v(x) , y = 1
! + 2"#c (z)
dz
0
x
$
into
Hc!(y) = "#y2 + pc (y) + c( )!(y) = $!(y) ,
where pc (y) = !"c (x) !
3#
2
$$"c !
#2 ("c$ (x)2
4 % + 2#" c (x)( ) .
For the smooth solitary wave !c (x) , we can obtain that pc (y) decays exponen-
tially at infinity, which gives the result that the operator Lc :H1 ! H "1 is self-adjoint
with essential spectrum c,![ ) and there are infinitely many eigenvalues which are
less than c . The function to the nth eigenvalue (in increasing order) have up to a con-
stant multiple, a unique eigenfunction with exactly (n ! 1) zeros. Referring to [15],
the Liouville transformation ensures that the same spectral information holds for the
operator Hc . Noting that (2.2) shows Hc (!cx ) = 0 . The property of the function !c
denotes that !cx has exactly one zero. The zero eigenvalue of Hc is simple, and
there is exactly one negative eigenvalue, while the rest of the spectrum is positive and
bounded away from zero.
Next we discuss orbital stability of the smooth solitary waves in Eq. (1.4). As we
known, a solitary wave is called orbital stable if a wave with an initial profile close to
the solitary wave remains close to some translate of it at all later times.
Definition 3.1. The solitary solution !c is orbital stable if for any ! > 0 , there
is ! > 0 such that if 0 < T ! " and u !C [0,T ); H1(R)( ) is a solution to (1.4) with
u0 ! "c H1 # $ ,
then
inf
!"R
u( # , t) $ %c ( # $ !) H1 & ! for every t ![0,T ) .
As for the above results, the stability depends on the convexity properties of the func-
tion d(c) = F(!c ) + cE(!c ) [15]. Then We give the following theorem.
Theorem 3.1. The solitary wave !c is stable if the function d(c) is strictly con-
vex, i.e., !!d (c) > 0 and unstable if the function d(c) is strictly concave, i.e.,
!!d (c) < 0 .
Next , we will prove the following theorem.
Theorem 3.2. The solitary wave solutions with c !R+ for Eq. (1.4) are stable.
Proof. Differentiating d(c) with respect to c , we get
1076 J. L. YIN
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
!d (c) = !F ("c ) + c !E ("c ),
#"c
#c
+ E("c ) = E("c ) .
Using (2.4) and the fact that !c is an even function, we have
!!d (c) = d
dc
"c2dx
R
# = d
dc
"c2dx
0
+$
# =
= d
dc
!"c"cx
2#"c + $
c ! "c
3
dx
0
+%
& =
= d
dc
!y 2"y + #
c ! y
3
dy
3c
0
$ =
= d
dc
y 2!y + "
c # y
3
dy
0
3c
$ .
Let y = 3cs . Then we have
!!d (c) = d
dc
9c2s 6"cs + #
c $ cs
ds
0
1
% =
= d
dc
9c3/2s 6!cs + "
1# s
ds
0
1
$ .
Let P(c) = 9c3/2s 6!cs + "
1# s
ds
0
1
$ . Clearly, P(c) is increasing at s in the inter-
val [0,1] , then !!d (c) > 0 . According to Theorem 3.1, we can finish the proof of
Theorem 3.2 and the solitary waves are stable for any wave speed.
4. Conclusions. The generalized KdV equation with combined dispersion has
smooth solitary wave solutions under the influence of the combined dispersion, which
can not be seen in Eq. (1.2) , (1.3) owing only nonlinear dispersion. By using qualita-
tive analysis method, the existence scope of smooth solitary wave was obtained. In the
existence scope, the solitary wave is orbit stable. Noticing the important role of the
combined dispersion, we will study other proposition of the combined dispersion in our
coming researches.
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– P. 564 – 567.
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Received 28.01.11
|
| id | umjimathkievua-article-2785 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:30:17Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/66/581025de2ea13500e5d52fa099afb766.pdf |
| spelling | umjimathkievua-article-27852020-03-18T19:36:38Z Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion Стійкість гладких відокремлених хвиль для узагальненого рівняння Кортевега –де Фріза з комбінованою дисперсією Yin, J. L. Інь, Дж Л. The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation. Розглянуто задачу про орбітальну стійкість гладких відокремлених хвиль для узагальненого рівняння Кортевега – де Фріза з комбінованою дисперсією. Отримані результати показують, що гладкі відокремлені хвилі є стійкими при довільній швидкості поширення хвиль. Institute of Mathematics, NAS of Ukraine 2011-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2785 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 8 (2011); 1071-1077 Український математичний журнал; Том 63 № 8 (2011); 1071-1077 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2785/2325 https://umj.imath.kiev.ua/index.php/umj/article/view/2785/2326 Copyright (c) 2011 Yin J. L. |
| spellingShingle | Yin, J. L. Інь, Дж Л. Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion |
| title | Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion |
| title_alt | Стійкість гладких відокремлених хвиль для узагальненого рівняння Кортевега –де Фріза з комбінованою дисперсією |
| title_full | Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion |
| title_fullStr | Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion |
| title_full_unstemmed | Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion |
| title_short | Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion |
| title_sort | stability of smooth sohtary waves for the generahzed korteweg - de vries equation with combmed dispersion |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2785 |
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