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 groups concentrated on linear and nonlinear ODEs and linear PDEs. The reduction in order of ODEs by Lie groups may differ from the reduction of the number of variables of linear and nonlinear PDEs. Lie, Marius Sophus. [Photograph]. Retrieved 7, 12, 2007, from Encyclopedia Britannica Online:

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.

Definitions

Review of ODE Hidden Symmetries Lie algebra of Type I hidden symmetries [U i,U j ] = c ij k U k No Type I hidden symmetries if c ij k = 0. Hidden symmetries if c ij k ≠ 0 and reduce by U i A.A. Both symmetries are nonlocal where A may be exponential and B may be linear. B. and Type I hidden symmetry

Example-Type I hidden symmetry ODE form with symmetry of U 1 and U 2 Use symmetry of U 1 : Use symmetry of U 2 : 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.

Contact symmetry reduces to Lie point symmetry Consider 2 symmetries of Reduce order by symmetry of G 4 where u=y, v=y’ and G 8, the contact symmetry, reduces to a local Lie symmetry. (contact symmetry)

Type II hidden symmetry reemerges as Lie point symmetry 3-D group generators: Reduce first by U 3 where u=x, v=xy’ then U 1 becomes non-local. Reduce next by G 2 to give the local generator

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.

2-D Burgers’ Equation Two-dimensional Burgers’ equation generalizes model equation for shocks and has 5 Lie symmetries Let Reduced Eq.-Target PDE then

Lie Symmetries of Reduced PDE Five Lie symmetries of which one is a Type II hidden symmetry Other PDEs reduce to target PDE where use reverse method to find other PDEs.

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

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 i, k =1, 2 Inherit 4-D+U ∞ Lie algebra, U ∞ = F(y 1,y 2 )∂/∂w

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.

Hidden Symmetries of PDE. 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 equation was derived from the general Einstein equations for the gravitational field by Plebanski. This nonlinear PDE has 4 independent 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. 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. G is a solution of the Monge-Amp è re Eq.

Provenance of Type II hidden symmetry Use invariants of Type II hidden symmetry given by generator The other PDE is then (not unique)

Useful inherited symmetries Commuting symmetries below do not lead to a useful symmetry if is used in the reduction of the PDE. Let p = t and q = u as reduction variables, then has no relevance for the reduced differential equation.

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, (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 (2006). 3. B. Abraham-Shrauner & K. S. Govinder,”Provenance of Type II hidden symmetries from nonlinear partial differential equations,” J. Nonl. Math. Phys. 13, (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.”