Vol. 1 No. 2
Sep. 1997
JOURNALOFSHANGHAIUNIVERSITY
Strange Attractor of KdV-Burgers Equation S h e n g P i n g x i n g (~-q- ~ ) (College of Sciences) Abstract The existence of isolated travelling wave solution for the KdV-Burgers equation is proved. It is shown that the strange attractor of the KdV-Burgers equation to isolated travelling wave is prevailing. The chaotic phenomena in numerical computation might provide a reasonable interpretation of turbulence. Key words: homoclinic orbit, isolated travelling wave, turbulence, honaeomorphism, chaotic phenomena. In this paper, the following KdV-Burgers equation in
&l+ &t
O2u _,73u
0
~-
u - z - - v _--2--;-+ / 3 - -
~
flw" - vw" + uw' + wu' = O, ww' = O.
\
one dimension is concerned:
~3
strange attractor, vector field
From the second equation it follows that w(x)--const.
: 0.
(1)
However u(x) is not a constant. It means that w--- O.
where v >0 and/7>0 are physical parameters and u(t,x) is
In order to discuss the stability of equilibria, we have to introduce a metric on C([0,oo)xR). This brings us a little
defined on
difficulty. We may consider a neighbourhood of u0 (x) by
[0,~) × R , . ~ c 3 ([0,oo) × R) c c ([0,oo)×R). Clearly one may first provide some initial conditions u(O,x)=Uo(X)~C(R ). Since we are interested in qualitative
analysis
of infinite-dimensional
dynamical
systems, specific initial conditions are of no concern at this moment. Later on ,we may focus on some subset o f C(R) as the set of initial functions. Let u satisfy the following steady-state equation:
cal .~--
dZu _ ~ u v-~-y+
o.
setting [u (x)-uo (x)]<~ for all x E R,u ~ N~ (Uo(X)). Sometimes such discdssion is inefficient. We may impose periodic conditions on u (t,x) in x. Then u (t,x) is restricted on C ([0,oo)x[0,w]). A norm in C [0,w] can be introduced by Ilu(.,x)lloo. The question on the stability of equilibria can be asked. Many different definitions of stability can be imposed, u(x) ~ SI~ is called stable if u (t,x)-->u (x) as t-->oo. The subset $1 of initial functions Uo(X) has the property that
u(t,x) with u(O,x)=uo(x) approaches to some equilibrium in SIe . Let
s,, : {
,,(,, x)
c 3 (co,
R)f.,
+uu,. - ~l,.,. + pu.~,..,.: 0 } is the set of equilibria or the set of steady-state solutions. It is quite clear that S,~, ;* ~b since all constants are in S,~. A topological or functional classification of S,e is extremely important. One property of S,~, can be easily obtained, namely any
be the collection of solutions. We may first consider a specific travelling wave
u( t,x )=u(x-ct) -- u( ~) with ~'=x-ct for some constant speed ¢.
s,':= {+- c,)
non-constant steady state solution u ~S, e
must be isolated. Suppose u ~ S,,~ is a non-constant steady-
,
d II
Let u = - - .
d~
state solution. For any given sufficiently-small e>0, we consider steady state solutions in the form o f u+~ w with w
> 0} s,,.
One has
blt=--Cld'~ btx=U', glxx =U"~Uxxx=U'.
being an arbitrary function. Substituting it into the equation, one obtains by looking at the order of • a n d ~ that
The KdV-Burgers equation turns out to be:
Received Feb. 16, 1997 Sheng Pingxing, Asso. Professor. College of Sciences, Shanghai University. 20 Chengzhong Road. Shanghai 201800
or
-cu' + uu' - vu" + flu" = 0
it 2
-cu + - - - ~t + flu" = A, 2
where A is an integral constant. Set u --v. One obtains the
~2
Journal of Shanghai Univetwity as a trajectory of initial value. By the uniqueness theorem
fbllowing polynomial autonomous system:
(2)
of initial value problem for autonomous systems, one can easily check ~i0,1] is indeed a homeomorphism called
If v=0 and fl=l, then the system (2) is Hamiltonian, which corresponds to KdV equation II t + l l l l x +11.~.~.~ ~ 0
%.](o) =
,
that first arose in the approximate theory of water waves.
u(t,x)=s(x-ct)
vector field homeomorphism t9"1o1. Let
with
then one has ~al0,®](C0) = ,
s(x)=3csech~- I - ~ - )
lira Cn =C~. n--).oo
Clearly all C,,'s are also simple closed curves by the is a solitary wave or a soliton. For the Hamiltonian system, the phase portrait can be easily figured out with (u~,O), (,2,0) being two equilibria, where u~.2=c +
bounded or unbounded. If C~ is bounded, it is equivalent to a w-limit set which is either a limit cycle or a limit homoclinic orbit rooting at (u2,0). If the first case happens,
+ 2A.
If ~ 0 , the system (2) is not Hamiltonian. Clearly (u~,0) and (u2,0) are still two equilibria with
we can consider Co containing C~ in its interior and sufficiently close to Coo, according to the stability of C~, and consider either ~P[03] ~ again or gJ[-i,0] r operating on Co.
u~=c + x ~ c2 + 2 A , u, = c -
uniqueness theorem of initial value problem. C~ is either
We then proceed until Coo or C_~o contains (u2,0). Hence
U + 2A•
if Coo is bounded, then Coo must be a limit homoclinic
namely uj, u2e SI~. We examine the linearization of the
orbit, Since v < < l , trajectories strongly oscillate around
system (2) at (u~, 0), (u2. 0) respectively. One can easily
(ul,0) and since vector field V travels to one side of
find that (u2.0) is always a saddle because
H(u,v)=d, and since algebraic system can not have unbounded generalized limit cycle other than the infinity l~11 C~ can not be the only infinity and can not be
+ ~ -
>0.
unbounded limit cycle. Therefore Coo must be bounded. (u,,0) is a unstable focus if v 2 < 4 f l ~ c "~+ 2A. Clearly if
Namely a limit homoclinic orbit exists. Therefore an
c2+2A<0 then u~ and u~ are complex. There are not any
isolated travelling wave of the KdV-Bua'gers equation exists. In numerical computation it is very difficult to find
• equilibria. Theorem 1
If v -~ <4fl-~c 2 + 2 A
and v < < l .
then
the system (2) has a linfit homoclinic orbit, namely the KdV-Burgers equation has an isolated travelling wave or a solitary wave. Proof: Under the assumption, (u~,0) is an unstable tbcus and (u2,0) is a saddle. Consider level ctu'ves 2 2,) 3 H(u30=-v/2+(Au+cu/--u/6)/fl=d.
invariant set A with the property that, given e >0, there is a set U of positive Lebesgue measure in the eneighbourhood of A such that x E U implies that the wlimit set o f x is contained in A • and the forward orbit o f x is contained in U. We shall call it an attractor strange if it
(H,,H,.)=((A+cu-u 2 / 2)/fl ,-v)
contains a transversal homoclinic orbit.
is the normal vector of level curves.
(u",v') • (El,,, H,.)=-v~/fl
such limit homoclinic orbit. It is ahnost impossible to achieve such trajectozy numerically. Chaotic phenomena occurs in computer simulation. Definition [sl An attractor is an indecomposable closed
The definition of a minimal
<_ O,
Ladyzenskaya (1987)
global
attractor by
and the definition of a global
namely vector field travels to one side of level curves H=d.
attractor by Hale (1988) or a universal attractor by Teman
Consider an initial simple closed curve Co containing (u~, 0)
(1988) for infinite dimensional dynamical systems can be
and sufficiently close to (ut,0). Since (u~,0) is an unstable tbcus, we define a vector field homeomol-phism
found in [6]. Theoreml~l
J'
.
~p[o.,].Co ---> C, = { :[~:=r(1)
•
r(O) = a ~ C o }
For any planar polynomial autonomous
system, there exists no unbounded limit set other than
Vol. 1 No. 2
Sep. 1997
Sheng P.: Strange Attractor of KdV-Burgers Equation
infinit2,. The Theorem 1 can only occur when v < < l . It matches with turbulence phenomena in experiments. When v<
93
strange and has a complicated structure. Such A~ other than As. is a good interpretation of turbulence. Another interesting question is thai as t gets large what will happen
(u~,O) is a strong focus, namely trajectories near (ub0)
for numerical scheme on PDE used in computing the KdV-
oscillate frequently.
Burgers equation starting near solitary wave? We are most
From the phase portrait of the system (2), one may see that zt~ e S,,~ is .asymptotically unstable. What kind initial
interested in directly showing the existence of non-trivial co- limit set for the KdV-Burgers equation (1) which is not
set B ~ C (R) will cause the travelling wave initiating at
a
some point Uo (x) e B to approach to some steady-state
dynamical systems can not be reduced to the system of
solution u" (x) ~S,,~ as t~oo or t ~ - o o ? The most
ODEs if one does not consider specific solutions like txavelling waves or if one even considers specific solutions.
exciting thing is that via the Theorem 1, one can find an initial set B c C (R) such that some travelling waves
steady-state
solution.
Most
infinite-dimensional
Do we know if there exists a non-trivial co-limit set which
starting in B will approach to the isolated travelling wave
is neither an equilibritnn nor a travelling wave? We are
represented by the limit homoclinic orbit which is not a
certainly interested in some bounded
~limit
set or
unbounded limit set other than infinity. If an ~ l i m i t set is
steady-state solution. Definition: A set A c S, is called a conditional attractor
bounded and non-trivial, then it is a bounded solution u"
of the KdV-Burgers equation if there exists a set B of
(t,x) either periodic in t or bounded in t or a union of
initial fhnctions such that the solution u(t, x) with u(0, x)=uo (x) e B will approach to A as t---> oo. A is called a
bounded solutions. If
conditional strange attractor of the KdV-Burgers equation
where zl" (s,x) is a non-trivial co-limit set, it is important to know that for almost all x~ R fixed, u (t,x) is infinitely
denoted by A, if A contains at least one isolated travelling wave solution and one steady-state solution. In numerical computation, any algorithm losses the stability near A,.
The
author
believes
that
u (t,x)--~u* (s,x)
as
t---~oo
oscillative as t--~oo in order to be a necessary condition for the existence of non-trivial co-limit set. For infinite-
chaotic
dimensional nonautonomous systeans, there are type-II
phenomena in computing due to As can be interpreted as
limit sets, namely a collection of limit points of trajectories
the turbulence of the KdV-Burgers equation. We know that
instead of only one trajectory.
the KdV-Burgers equation only approximates the reality in
u~(x) with u (t, r , x)---~ u~ (x) as t --~oo for r e A where A
physics or other subjects. An analytic solution may not be the exact solution of the reality. Numerical solutions of the KdV-Burgers
equation
also
approximates
the
exact
solutions of the equation. Therefore, numerical solutions ,nay be good approximation of the real problem. That is why we use chaotic phenomena in computing due to A, to represent the turbulence of the KdV-Burgers equation. For an n-dimensional autonomous system the solution of an initial value problem is unique. Therefore, for any autonomous system, no chaotic phenomena can occur in analytic solutions. It should also be emphasized that nonautonomous systems can not be transformed to autonomous systems by raising one dimension higher when one discusses limit sets of non-autonomous systems (for detailed discussion see [8]). We may ask why A, is a conditional strange attractor of the KdV-Burgers equation? Although A~ contains a limit homoclinic orbit, A~ is welldefined. A really named conditional strange attractor should be A.~* due to A~ in numerical computation.
A* is
Do we have a collection of
is a set of parameter which possesses the following property that ur (x) is a periodic solution for r e[0,co]=A, namely u r (x)=u (x,x) is a solution of the KdV-Burgers equation?
From
above
necessary
condition
for the
existence of nontrivial co-limit set, one can easily have the following proposition. Proposition: Along any solution u(t,x)=fp(x) there exists no non-trivial to-limit set.
T(t),
Proof: Suppose u(t,x)=ep (x) ~ (t) is a solution of the KdV-Burgers equation, one has
~o(x) ~ (t)= -q, (x)~ (x). ~ (t) + v ~ (x) ~(t) -/~ ~," (x) ~'(0 or for almost all x~ R fixed, ¢ ( , ) = - ~ o ( x ) ~ , : ( , ) + v~ "(x)-a~"(x). --~,~,t,j. ~o(x) For each fixed x, 7-' (t) monotonically approaches to To=const. as t ~oo. Hence u(t,x)=¢o ( x ) ~ (t) can not infinitely oscillate as t---~oo. It implies that u(t,x)=~o (x) ~(t)
Journal of Shanghai University
94 can not approach to nontrivial co-limit set of the KdVBurgers equation. We are further interested in knowing if there is any solution u(t,x), which is not travelling wave and which can not be represented by u(t,x)=~x) ~ (t), approaching to a non-trivial co-limit set which is not the isolated travelling wave solution for v < < l .
References 1 FaddeevL. and Zahharov V., Korteweg-de Vries equation as a completely integrable Hamiltonian system. ,Z Math. Phys'., 12, 1548-1551. (1971) 2 Lax P., Integrals of nonlinear equations of evolution and solitary waves. Comm. Pm'e Appl. Math., 21,467-490. (1968) 3 Dodd R. K. et aL Solitons and nonlinear wave equations, Academic Press. 1982 4 BergerM. S., NonlmeartO, andfimctional analysis', Lectm'es on nonlinear problems in mathematical analysis, Academic Press. 1982
5 Guckenheimer J. and Holmes P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Applied Mathematical Sciences, 42. (1983) 6 Hale J. K. and Raugel G., Attractors for dissipative evolutionary equations, Report of Center for Dynamical Systems and Nonlinear Studies. G. I. T. 1992 7 Teman R., Infinite dimensional dynamical systems m mechanics and physics, Springer-Verlag. 1988 8 Sheng P. X., Chaos in dynamical systems, preprint at conununication. 1992 9 Sheng P. X., A characteristic of dynamical systems in R: Huanghuai J. 11(2), 46-50. 1995 10 Sheng P. X., A class of predator-prey systems with ratiodependence. Comm. On AppL Math. and Comp, 9(1), 71-75. 1995 11 ShengP. X. and Hastings S. P., A generalization of PoincareBendixson theorem (preprint). 1996 12 Li D. Q. and Chen Y. M., Nonlinear evohttional y equatiom', Chinese Science Press. 1989 13 Ye Q. X. and Li Z. Y., Introduction to reaction-diffitsion eqnations, Chinese Science Press. 1994