Brookings Hall at Washington University

Type II hidden symmetries of nonlinear partial differential equations

Barbara Abraham-Shrauner &

Keshlan S. Govinder#

#University of KwaZulu-Natal, Durban 4041, South Africa.

Marius Sophus Lie

Sophus Lie’s development of transformation groupsconcentrated on linear and nonlinear ODEs and linearPDEs. The reduction in order of ODEs by Lie groupsmay differ from the reduction of the number ofvariables of linear and nonlinear PDEs.

Lie, Marius Sophus. [Photograph]. Retrieved 7, 12,2007, from Encyclopedia Britannica Online: http://www.britannica.com/eb/art-12613

Motivation

• Predict reduction path for a differential equation from the Lie algebra of the Lie point group symmetries.• Reduction of order of ODEs• Reduction of number of variables of PDEs

• Useful for symbolic computation of Lie point group symmetries to know the predicted reduction paths.

• Little systematic investigation reported of Type I and II hidden symmetries of PDEs before our papers.

• If reduction path for PDEs is insufficient, are non Lie point group symmetries needed?• For example: contact, Lie-Backlund, conditional,generalized, nonlocal, nonclassical, etc.

Significant Finding

Useful Lie point symmetries that occur in a reduction path differ for ODEs & PDEs

• Inheritance of Lie symmetries is predicted for ODEs from Lie algebra of original ODE except may need Lie algebra of reduced ODE if nonlocal or contact symmetries exist.

• Inheritance of useful Lie symmetries of PDEs may not include some symmetries retained for ODEs.

• Appearance of new Lie point group symmetries differ in reduction paths of ODEs and PDEs.

• Definition: Type I (II) hidden symmetries of ODEs are Lie point symmetries that disappear (appear) upon reduction of order of the ODE by one by using another Lie symmetry.

• Definition: Type I (II) hidden symmetries of

PDEs are Lie point symmetries that disappear (appear) upon reduction of the number of variables by one of the PDE by using another Lie symmetry.

Definitions

Review of ODE Hidden Symmetries

Lie algebra of Type I hidden symmetries

[Ui,Uj] = cijk Uk

No Type I hidden symmetries if cijk = 0.

Hidden symmetries if cijk ≠ 0 and reduce by Ui

€

Ui≠U jA.

Both symmetries are nonlocal where A may be exponential and B may be linear.

B.

€

Ui≠U j≠Uk€

U j=Ukand

Type I hidden symmetry

Example-Type I hidden symmetry

€

U1= ∂∂x ,

€

[U1,U2]=U1

ODE form with symmetry of U1and U2€

U2 =x ∂∂x

€

F(y, ′ ′ y ′ y 2

, ′ ′ ′ y ′ y 3

)=0

€

u= ′ y ,v=y,

€

U2 →u ∂∂u

Use symmetry of U1:

Use symmetry of U2 :

€

u=x ′ y ,v=y,

€

U1→uexp( dyu∫ ) ∂

∂u

is nonlocal (exponential).

Type II Hidden symmetries

• Type II hidden symmetry may result from contact symmetry of ODE by reduction of order of ODE by Lie point symmetry.

• Type II hidden symmetry may result from reduction of order of ODE by two Lie point symmetries where case B applies with and

• Symmetries of then used where intermediate of is nonlocal.

€

[Ui,U j]=ckijU

k

€

Ui≠U j≠Uk

€

Ui

€

Uk

€

U j

Contact symmetry reduces toLie point symmetry

Consider 2 symmetries of

€

′ ′ ′ y =0

€

G4 = ∂∂x ,

Reduce order by symmetry of G4

€

G8 →v2

2∂∂u

where u=y, v=y’ and G8, the contact symmetry, reduces to a local Lie symmetry.

€

G8 = ′ y ∂∂x+ ′ y 2

2∂∂y (contact symmetry)

Type II hidden symmetry reemerges as Lie point symmetry

3-D group generators:

€

U1= ∂∂x ,

€

U2 = ∂∂y,

€

U3 =x ∂∂y

Reduce first by U3 where u=x, v=xy’ thenU1 becomes non-local.

€

U1→ ∂∂u+(v

u+ vu2∫ du) ∂

∂v=G1,

€

U2 →− ∂∂v=G

2

Reduce next by G2 to give the local generator

€

G1→ ∂∂z+w

z∂

∂w!

Lie Symmetries of PDEs in reduction path

• Appearance of new Lie point symmetries• Not from contact or nonlocal symmetries as do

not reduce the order of PDEs.• Provenance to be discussed.

• Disappearance of useful Lie point symmetries• Normal subgroup of Lie symmetry may none the less not be a useful symmetry when reduce thenumber of variables. • Applies to zero commutators of generators.

Provenance of Type II hidden symmetries of PDE

• Type II hidden symmetries of a PDE are new symmetries not inherited from the original PDE but from other PDEs that reduce to the same target PDE.

• The reduction in the number of variables is assumed to be by the same Lie symmetry for the original PDE and the other PDEs where the other PDEs have the same independent and dependent variables.

PDEs with Type II hidden symmetries

• 3-D (spatial) wave equation (Stephani )

• 2-D (spatial) wave equation• 2-D (spatial) Burgers’ equation (Broadbridge)

• Second heavenly equation

• Model nonlinear PDEs

• Other examples are possible

Earth’s Bow Shock is Three-dimensional

Burgers’ equation was a1-D (spatial) model for a shock wave but real shocks are frequently 2-D or 3-D. Spreiter’s model for the Earth’s bow shock shape was based on gas dynamics for an axisymmetric projectile and magnetic fields were neglected.

http://sci.esa.int

2-D Burgers’ Equation

• Two-dimensional Burgers’ equation generalizes model equation for shocks and has 5 Lie symmetries

€

ut +uux=uxx+uzz

€

U1= ∂∂x,

€

U2 = ∂∂z,

€

U4 =t ∂∂z+ ∂

∂u,

€

U5 =2t ∂∂t +x ∂

∂x+z ∂∂z−u ∂

∂u.

€

U3 = ∂∂z,

€

Ua =aU1+U2

€

u=w(t,ρ ),

€

ρ=z−xaLet

Reduced Eq.-Target PDE

€

wt +wwρ =1+a2

a2 wρρ

then

Lie Symmetries of Reduced PDE

• Five Lie symmetries of which one is a Type II hidden symmetry

€

U5=t2 ∂

∂t+tρ ∂

∂ρ+(ρ−tw) ∂

∂w.

• Other PDEs reduce to target PDE where use

reverse method to find other PDEs.

€

ut +uux=1+a2

a2uzz+D(uxx+uzz

a)(uxx+2uxz

a+uzz

a2)

€

D=constant

Two-dimensional (spatial)Wave

Partial Differential Equations Higher-dimensional PDE: Vibrating rectangular membranes and nodes.Anton DzhamayDepartment of MathematicsThe University of MichiganAnn Arbor, MI 48109

uxx+uyy−utt=0.

Linear 2-D wave Eq. is invariant under Lie group with 11-D + U∞ Lie algebra. U∞ = F(x,y,t)∂/∂u. Equation in cylindrical coordinates.

Hidden Symmetries 2-D wave Eq.

Reduce PDE by scaling symmetry to

urr+ur /r+uθθ /r2−utt=0.

i, k =1, 2∂2w∂yi∂yk

(δik−yiyk)−2yi∂w∂yi

=0.

€

y1 = x/t,

Inherit 4-D+U∞ Lie algebra, U∞= F(y1,y2)∂/∂w€

y2 = y/t,

€

w=u.

Hidden Symmetries-2-D wave Eq

Reduce PDE by axial rotation to ODE

This ODE has the full 8-D Lie algebra with three inherited symmetries from the 2-D wave equation. Are any of the other five Lie symmetries inherited from the intermediate PDE? That question is discussed on the next slide.

v=(x2+y2)/t24v(1−v)wvv+4(1−1.5v)wv=0.

Hidden Symmetries of PDE.

€

R2

(1−R2

)wRR+R(1−2R2

)wR+wθθ =0.

R=r /t.

• PDE has 4 Lie symmetries +U∞ inherited plus three Type II hidden symmetries. • Reduced ODE has an inherited Type II and regular symmetry. U∞ becomes two Lie symmetries • Other 4 Lie point symmetries are Type II hidden symmetries. • Extra Lie symmetries of the PDE were found by classical method and reverse method.

Second Heavenly Equation

The second heavenly equationwas derived from the generalEinstein equations for thegravitational field by Plebanski.This nonlinear PDE has 4independent variables & invariant under an infinite Lie algebra.

J. Lemmerich: Max Born, James Franck, der Luxus des Gewissens..., Ausstellungskatalog, Staatsbibliothek Preussischer Kulturbesitz, 1982 einst_1.jpg A.E. in his study in Haberlandstr., p. 79

Type II hidden symmetries of 2nd Heavenly equation

Second Heavenly Eq.

€

θxxθyy−θxy2

+θxw+θyz =0.

This nonlinear PDE is invariant under an infinite Lie group. Reduce by translations in w, find another infinite Lie group. A Type II hidden symmetry results in a new solution.

€

u(x,y,z)= 1z

G(xz ,y z)−x2y

4z.

G is a solution of the Monge-Ampère Eq.

€

GrrGss −Grs2 =0.

Provenance of Type II hidden symmetry

• Use invariants of Type II hidden symmetry given by

generator

• The other PDE is then (not unique)

€

uxxuyy −uxy2 +uyz +

uxwuyy=0.

€

V12

=(2uz+x2y) ∂∂u

−4xz ∂∂x

+2yz ∂∂y

−4z2 ∂∂z

.

Useful inherited symmetries

Commuting symmetries below do not lead to a useful symmetry if is used in the reduction of the PDE.

€

G1= ∂

∂x,

€

G2=θ(t,u) ∂

∂x.

€

G1

Let p = t and q = u as reduction variables, then has no relevance for the reduced differential equation.

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G2

Conclusions• Type II hidden symmetries of PDEs can yield other solutions than

predicted from original Lie algebra.• Provenance of Type II hidden symmetries has been shown to be Lie point

symmetries from other PDEs.• Reverse method for finding provenance of Type II hidden symmetries is

more complete than ad hoc approach. • Lie symmetry of normal subgroup may not be useful Lie point symmetry

of reduced PDE.• Prediction of useful Lie symmetries from Lie algebra of original PDE is

an open question.

Some of our recent relevant references

1. K. S. Govinder, “Lie Subalgebras, Reduction of Order and Group Invariant Solutions,” J. Math. Anal. Appl. 250, 720-732 (2001).2. B. Abraham-Shrauner, K. S. Govinder & D. J. Arrigo, “Type-II hidden symmetries of the linear 2D and 3D wave equations,” J. Phys.A: Math. Gen. 29 5739-47 (2006).3. B. Abraham-Shrauner & K. S. Govinder,”Provenance of Type II hidden symmetries from nonlinear partial differential equations,” J. Nonl. Math. Phys. 13, 612-622 (2006).4. B. Abraham-Shrauner, “Type II hidden symmetries of the second heavenly equation,”Phys. Lett. A (in press).5. B. Abraham-Shrauner & K. S. Govinder, “Master Partial Differential Equations for a

Type II Hidden Symmetry.”