Symbolic Computation of
Conservation Laws of
Nonlinear PDEs
Douglas Poole and Willy Hereman
Department of Mathematical and Computer Sciences
Colorado School of Mines
Golden, Colorado, U.S.A.
lpoole@mines.edu
whereman@mines.edu
Joint AMS-MAA Meetings
Thursday, January 14, 2010
Acknowledgements
Research supported in part by NSF
under Grant CCF-083783
Outline
•
Overview of conservation laws
•
Conservation laws for the Zakharov-Kuznetsov
equation
•
Presentation of ConservationLawsMD.m, a
Mathematica program to compute conservation
laws
•
Conservation laws for the Manakov-Santini System
Overview of Conservation Laws
•
The computation of conservation laws was an
integral part of an in-depth study to find solutions
to the Korteweg-de Vries (KdV) equation
•
An infinite set of conservations laws is a factor
determining that the PDE is completely integrable
•
In a system described by a PDE, conservation laws
I
show which physical quantities are conserved
I
lead to discoveries toward finding solutions of
PDEs such as the Inverse Scattering Transform
I
aid in the study of qualitative properties of a
PDE
I
aid in the design of numerical solvers
Conservation Law
Dt ρ + Div J = 0
ρ is the “conserved density”
on PDE
J is the “flux”
The continuity equation is satisfied for all solutions of
the PDE.
In 1-D, J = J and Div J = Dx J
In 2-D, J = (J 1 , J 2 ) and Div J = Dx J 1 + Dy J 2
In 3-D, J = (J 1 , J 2 , J 3 ) and Div J = Dx J 1 + Dy J 2 + Dz J 3
The Zakharov-Kuznetsov (ZK) Equation
and Conservation Laws
The ZK equation models ion-sound solitons in a low
pressure uniform magnetized plasma
The (2+1)-dimensional ZK equation:
ut + αuux + β(u2x + u2y )x = 0
Conservation Laws:
Dt u + Dx α2 u2 + βu2x + Dy βuxy = 0
3
2
2
Dt u2 + Dx 2α
u
−
β(u
−
u
x
y ) + 2βu(u2x + u2y )
3
− Dy 2βux uy = 0
More Conservation Laws:
2
2
2
2
2
3α 4
+
u
)
+
u
Dt u3 − 3β
(u
+
D
u
+
3βu
u
−
6βu(u
x 4
2x
x
y
x
y)
α
2
2
+ 3βα (u22x − u22y ) − 6βα (ux (u3x + ux2y ) + uy (u2xy + u3y ))
2
+ Dy 3βu2 uxy + 6βα uxy (u2x + u2y ) = 0
3
2
2
Dt tu2 − α2 xu + Dx t( 2α
u
−
β(u
−
u
x
y ) + 2βu(u2x + u2y ))
3
2β
1
u
)
+
u
−
D
2β(tu
u
+
xuxy ) = 0
− x(u2 + 2β
x
y
x
y
2x
α
α
α
COMPUTER DEMONSTRATION
The Generalized Zakharov-Kuznetsov Equation
and Conservation Laws
ut + αun ux + β(u2x + u2y )x = 0,
where n is rational, n 6= 0
Conservation Laws:
Dt u + Dx α2 u2 + βu2x + Dy βuxy = 0
3
2
2
Dt u2 + Dx 2α
u
−
β(u
−
u
x
y ) + 2βu(u2x + u2y )
3
− Dy 2βux uy = 0
The third conservation law:
2
2
+
u
Dt un+2 − (n+1)(n+2)β
(u
x
y)
2α
2(n+1)
n+1
+ Dx (n+2)α
u
+
(n
+
2)βu
u2x
2(n+1)
n
− (n + 1)(n + 2)βu
−
(n+1)(n+2)β 2
α
(u2x
+
u2y )
+
(n+1)(n+2)β 2
2
(u
2x
2α
(ux (u3x + ux2y ) + uy (u2xy + u3y ))
+ Dy (n + 2)βun+1 uxy +
(n+1)(n+2)β 2
α
− u22y )
uxy (u2x + u2y ) = 0.
The Manakov-Santini System
and Conservation Laws
utx + u2y + (uux )x + vx uxy − u2x vy = 0
vtx + v2y + uv2x + vx vxy − vy v2x = 0
Conservation Laws:
Dt f ux vx + Dx f (uux vx − ux vx vy − uy vy )
− f 0 y(ut + uux − ux vy ) + Dy f (ux vy + uy vx + ux vx2 )
+ f 0 (u − yuy − yux vx ) = 0,
where f = f (t) is arbitrary
Dt f (2u + vx2 − yux vx ) + Dx f (u2 + uvx2 + uy v
− vy2 − vx2 vy − y(uux vx − ux vx vy − uy vy ))
− f 0 y(vt + uvx − vx vy ) − (2f x − f 0 )y 2 (ut + uux − ux vy )
− Dy f (ux v − 2vx vy − vx3 + y(ux vx2 + ux vy + uy vx ))
− f 0 (v − y(2u + vy + vx2 )) + (2f x − f 0 y 2 )(ux vx + uy ) = 0,
where f = f (t) is arbitrary
plus 3 others
Conclusions and Future Work
•
The conservation laws program is fast and has
computed conservation laws for a variety of
PDEs. Improvements to the code will allow for a
broader class of PDEs.
•
Future research will include a study of integrability
of DDEs, IDEs and delay DEs, leading to
techniques, algorithms, and software to compute
conservation laws, symmetries, and recursion
operators.
•
Software can be found at
http://inside.mines.edu/∼whereman