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Applications of the package CRACK tosimplify large systems

Thomas Wolf

Department of Mathematics,Brock University

St.Catharines, Ontario, Canada

June 25, 2009

Outline

Poisson Structures

Supersymmetric PDEs

The Computer Algebra Package CRACK

Poisson Structures (with A. Odesskii)

To define a Poisson structure one needs to specify:

I a set of variables, say,x1, ..., xn

I so-called Poisson brackets for each pair of variables:

{xi , xj} = ωij(x1, ..., xn).

Here ωij are functions in x1, ..., xn.

Example:Let the set of variables be p, q and Poisson brackets be

{p, p} = {q, q} = 0, {p, q} = 1, {q, p} = −1.

To define a Poisson structure one needs to specify:I a set of variables, say,

x1, ..., xn

{xi , xj} = ωij(x1, ..., xn).

{p, p} = {q, q} = 0, {p, q} = 1, {q, p} = −1.

x1, ..., xn

{xi , xj} = ωij(x1, ..., xn).

{p, p} = {q, q} = 0, {p, q} = 1, {q, p} = −1.

x1, ..., xn

{xi , xj} = ωij(x1, ..., xn).

{p, p} = {q, q} = 0, {p, q} = 1, {q, p} = −1.

Poisson structuresGiven a Poisson structure we define Poisson brackets for anyfunctions f , g in x1, ..., xn by

{f , g} =n∑

∂f∂xi

∂g∂xj

{xi , xj}

∂f∂xi

∂g∂xj

Functions ωij should satisfy the following constrains:I Anti-symmetry: {f , g} = −{g, f} for all f , g. This means

that ωji = −ωij .I Jacobi identity: {f , {g, h}}+ {g, {h, f}}+ {h, {f , g}} = 0 for

all f , g, h.

Formulating these conditions for f , g, h = x1, .., xn gives acomplete set of conditions for the ωij : an overdeterminednon-linear system of PDEs.

{f , g} =n∑

∂f∂xi

∂g∂xj

{xi , xj}

∂f∂xi

∂g∂xj

all f , g, h.

{f , g} =n∑

∂f∂xi

∂g∂xj

{xi , xj}

∂f∂xi

∂g∂xj

that ωji = −ωij .

I Jacobi identity: {f , {g, h}}+ {g, {h, f}}+ {h, {f , g}} = 0 forall f , g, h.

{f , g} =n∑

∂f∂xi

∂g∂xj

{xi , xj}

∂f∂xi

∂g∂xj

all f , g, h.

{f , g} =n∑

∂f∂xi

∂g∂xj

{xi , xj}

∂f∂xi

∂g∂xj

all f , g, h.

Applications of Poisson structures IHamiltonian systems: To define a Hamiltonian system oneneeds to specify a Poisson structure and an additional functionH(x1, ..., xn) called Hamiltonian.

The dynamics is given by:

dt= {H, xi}

In this case for any function f (x1, ..., xn) we also have

= {H, f}.

A system is called integrable if there exist sufficiently manyfunctions H0(= H), H1, ... such that {Hi , Hj} = 0. In particularwe have

dt= {H, Hi} = 0

so Hi = const .

Applications of Poisson structures IHamiltonian systems: To define a Hamiltonian system oneneeds to specify a Poisson structure and an additional functionH(x1, ..., xn) called Hamiltonian.The dynamics is given by:

dt= {H, xi}

= {H, f}.

dt= {H, Hi} = 0

so Hi = const .

Applications of Poisson structures IHamiltonian systems: To define a Hamiltonian system oneneeds to specify a Poisson structure and an additional functionH(x1, ..., xn) called Hamiltonian.The dynamics is given by:

dt= {H, xi}

= {H, f}.

dt= {H, Hi} = 0

so Hi = const .

Applications of Poisson structures II

In pure mathematics many important objects carry a naturalPoisson structure. It is also interesting to study the geometry ofso-called symplectic leaves connected with each Poissonstructure.

Examples of Poisson structuresI Let {xi , xj} = cij . This is a Poisson structure if cji = −cij .

The so-called canonical Poisson structure with variablesp1, ..., pn, q1, ..., qn and brackets

{pi , pj} = {qi , qj} = 0, {pi , qj} = δij

belongs to this type.

I Linear Poisson brackets:

{xi , xj} =n∑

ckij xk .

The Jacobi identity gives an overdetermined system ofquadratic algebraic equations for ck

ij . These equations

mean that ckij are structure constants of a Lie algebra.

I Quadratic Poisson brackets:

{xi , xj} =n∑

k ,l=1

cklij xkxl .

There are two type of such structures: rational and elliptic.

belongs to this type.I Linear Poisson brackets:

{xi , xj} =n∑

ckij xk .

{xi , xj} =n∑

k ,l=1

cklij xkxl .

belongs to this type.I Linear Poisson brackets:

{xi , xj} =n∑

ckij xk .

{xi , xj} =n∑

k ,l=1

cklij xkxl .

Elliptic Poisson structuresElliptic Poisson structures have the following applications:

I The theory of quantum groups.

I Non-commutative geometry. Elliptic algebras and theirquotients are examples of non-commutative algebraicmanifolds.

I Cohomologies of associative algebras. Elliptic algebrasare examples of Kozul algebras.

I Integrable systems. Elliptic algebras are a good tool instudying integrable systems connected to solutions of theYang-Baxter equation.

The most interesting known examples of these structuressatisfy the following property: there exists a group ofautomorphysms of the form:

xi → εixi , xi → xi+1

where ε is an nth root of unity and indices are given modulo n.

I The theory of quantum groups.I Non-commutative geometry. Elliptic algebras and their

quotients are examples of non-commutative algebraicmanifolds.

Quadratic (Z/3Z)2 invariant Poisson structures

The Poisson structures to be investigated have 9 dynamicvariables xij , i , j = 0, 1, 2 and Poisson brackets

{xij , xkl} =∑

p,q,r ,s

aijklpqrsxpqxrs. (1)

We assume the following symmetries:

xij → x(i+1)j , xij → xi(j+1), xij → εixij , xij → εjxij

This means that apart from the usual symmetries

aijklpqrs = aijklrspq = −aklijpqrs

the coefficients aijklpqrs also obey the restriction that aijklpqrs = 0if i + k 6= p + r mod 3 or j + l 6= q + s mod 3.

Quadratic (Z/3Z)2 invariant Poisson structures

The Poisson structures to be investigated have 9 dynamicvariables xij , i , j = 0, 1, 2 and Poisson brackets

{xij , xkl} =∑

p,q,r ,s

aijklpqrsxpqxrs. (1)

We assume the following symmetries:

xij → x(i+1)j , xij → xi(j+1), xij → εixij , xij → εjxij

This means that apart from the usual symmetries

aijklpqrs = aijklrspq = −aklijpqrs

the coefficients aijklpqrs also obey the restriction that aijklpqrs = 0if i + k 6= p + r mod 3 or j + l 6= q + s mod 3.

Computational ProblemThe Jacobi identities

{xij , {xkl , xmn}}+ {xkl , {xmn, xij}}+ {xmn, {xij , xkl}} = 0

are formulated using the Poisson structures

{xij , xkl} =∑

p,q,r ,s

aijklpqrsxpqxrs.

These identities have to be satisfied identically in the dynamicalvariables xij . Therefore all coefficients of different powers ofdifferent xij are set to zero resulting in 2754 quadratic equationsmost of which are redundant leaving 62 equations with 354terms for 20 unknowns aijklpqrs.

We demonstrate the subcase that two of the a’s are zero:

a00100112 = a00011021 = 0

which is the simplest of 5 sub-cases that have to be considered.

{xij , xkl} =∑

p,q,r ,s

aijklpqrsxpqxrs.

a00100112 = a00011021 = 0

{xij , xkl} =∑

p,q,r ,s

aijklpqrsxpqxrs.

a00100112 = a00011021 = 0

Outline

Poisson Structures

Supersymmetric PDEs

About the Problem I (work done with A. Kiselev)

Starting with a program SSTOOLS for doing supersymmetricalgebra and calculus an online database of supersymmetricevolutionary PDE of higher order symmetries was created.When analysing some classes of the 10,000 web pagedatabase Kiselev and TW found a N=2 supersymmetricgeneralization of the Burger’s equation.

The computation to be discussed deals with the geometricproblem of constructing Gardner’s deformation for the BosonicLimit of this equation. Invented in late 60’s, Gardner’s integrabledeformations of PDE consist of parametric extensions, subjectto further restrictions, for the equations under study, combinedwith differential substitutions back to the original system.

About the Problem I (work done with A. Kiselev)

Starting with a program SSTOOLS for doing supersymmetricalgebra and calculus an online database of supersymmetricevolutionary PDE of higher order symmetries was created.When analysing some classes of the 10,000 web pagedatabase Kiselev and TW found a N=2 supersymmetricgeneralization of the Burger’s equation.

The computation to be discussed deals with the geometricproblem of constructing Gardner’s deformation for the BosonicLimit of this equation. Invented in late 60’s, Gardner’s integrabledeformations of PDE consist of parametric extensions, subjectto further restrictions, for the equations under study, combinedwith differential substitutions back to the original system.

About the Problem II

For the Bosonic Limit at hand, we impose the assumption thatboth the extension and the substitution be polynomial andweight-homogeneous. This yields the algebraic system that islinear in the undetermined coefficients responsible for theextension and that is quadratic in the constants contained in thebackward substitution.

Six solutions, grouped in two classes, are obtained. Thecorresponding Gardner deformations are geometricallynontrivial. However, Kiselev proved that none of the sixsolutions can be lifted from the Bosonic Limit to the full N=2supersymmetric system.

About the Problem II

For the Bosonic Limit at hand, we impose the assumption thatboth the extension and the substitution be polynomial andweight-homogeneous. This yields the algebraic system that islinear in the undetermined coefficients responsible for theextension and that is quadratic in the constants contained in thebackward substitution.

Six solutions, grouped in two classes, are obtained. Thecorresponding Gardner deformations are geometricallynontrivial. However, Kiselev proved that none of the sixsolutions can be lifted from the Bosonic Limit to the full N=2supersymmetric system.

About the Computation I

I A Groebner basis computation is too large.

I Substitutions? Yes as problem is linear in some variables.I But substitution lead to blow-up of the system in size and

thus in time too.I A slower growth but still blow-up happens if

case-generating substitutions are allowed (each casedistinction may double the computation time of theremaining computatin of that case)

I Shortenings are essential as they provide short equationsin which fewer substitutions are done and whichsubsequently grow slower, but more importantly shorteningproduces shorter substitution equations

I A Groebner basis computation is too large.I Substitutions? Yes as problem is linear in some variables.

I But substitution lead to blow-up of the system in size andthus in time too.

I A slower growth but still blow-up happens ifcase-generating substitutions are allowed (each casedistinction may double the computation time of theremaining computatin of that case)

I A Groebner basis computation is too large.I Substitutions? Yes as problem is linear in some variables.I But substitution lead to blow-up of the system in size and

thus in time too.

I A slower growth but still blow-up happens ifcase-generating substitutions are allowed (each casedistinction may double the computation time of theremaining computatin of that case)

About the Computation II

I Experience: After each substitution many shorteningsbecome possible.

I Giving shortenings a strictly higher priority than longsubstitutions is a very safe but very slow solving strategy.

I Controlled speedup: Partially interactively do a fewsubstitutions at the time and then longer periods ofshortenings.

I How can such an interactive computation be remembered?→ by a backup of all interactive commands doneautomatically.

I In this comutation soon many equations factorize and onlycase-generating substitutions become possible.

About the Computation III

I Although CRACK has elaborate heuristics about whichcase distinctions to do first, CRACK does of course notknow about the intrinsic (geometric) relevance of allpossible different case distinctions one could try.

I How to find out which cases are cricial and which not?I By running a computation where case distinctions have a

high priority, and by inspecting case distinctions where atleast one case was solved quickly.

I Both cases .. = 0 and .. 6= 0 are both very useful (if the lhsis simple and occurs frequently as factor or as coefficient).Thus, if one of both cases is a strong restriction whichleads to contradictions that no solution exists in that case(typically performed quickly) then one got a goodsimplification of the other (general) case for a relativelysmall price.

I How to find out which cases are cricial and which not?

I By running a computation where case distinctions have ahigh priority, and by inspecting case distinctions where atleast one case was solved quickly.

About the Computation IVI Example: This is a list of case distinctions produced after a

CRACK run but one can print a list of cases also during aCRACK run: (see emacs file).

I In this (labour intensive) way the following (probablyoptimal) sequence of case distinctions lead to the fullsolution:

case extra assumption # of solutions1. p1 3 = 0 02. p1 3 6= 32.1. p1 8 = 02.1.1. p2 302 − 1 = 0 02.1.2. p2 302 − 1 6= 02.1.2.1. p1 6 = 0 42.1.2.2. p1 6 6= 02.1.2.2.1. p1 6 + 4 = 0 02.1.2.2.2. p1 6 + 4 6= 0 02.2. p1 8 6= 0 2

Outline

Poisson Structures

Supersymmetric PDEs

Purpose of CRACK

I a computer algebra package written in REDUCE for thesolution of large over-determined systems of algebraic,ordinary or partial differential equations

I especially suited for large over-determined systems withpolynomial non-linearity

Purpose of CRACK

I a computer algebra package written in REDUCE for thesolution of large over-determined systems of algebraic,ordinary or partial differential equations

I especially suited for large over-determined systems withpolynomial non-linearity

History

I initially designed for the automatic integration ofover-determined linear PDE systems (resulting inLie-symmetry and conservation law computations ofdifferential equations)

I later emphasis on interactive access for solving PDEs(resulting in same type of investigations but higher orderPDEs)

I which together with a better handling of non-linearityenabled to treat large algebraic systems (resulting in theclassification of integrable Hamiltonians),

I various extensions in last years.

History

Handling large Systems

I a module to reduce the length of equations and by thatmake the system more sparse,

I the ability to compute a bound on the size of the systemafter doing a specific substitution without doing thesubstitution,

I the option to do substitutions “bottom up”, i.e. only onesubstitution in the shortest possible equation using only ashorter equation,

I the possibility to tackle systems that are too large even tobe formulated, by having a buffer of equations which isdynamically emptied (equations are read from it into thecomputation) and filled (new equations are generated assolution progresses).

Example for Shortening

0 = x + +A(z, ..) (2)

0 = y + A(z, ..) (3)

total ordering: x > y > z > ..A is a large expression.GB: -

Shortening:

0 = x − y (4)

0 = y + A (5)

Obvious benefit:I In further steps growth comes only from one A in (5) and

not from two A in (2) and (3).

0 = x + +A(z, ..) (2)

0 = y + A(z, ..) (3)

total ordering: x > y > z > ..A is a large expression.GB: -Shortening:

0 = x − y (4)

0 = y + A (5)

0 = x + +A(z, ..) (2)

0 = y + A(z, ..) (3)

total ordering: x > y > z > ..A is a large expression.GB: -Shortening:

0 = x − y (4)

0 = y + A (5)

Non-obvious benefit:I reinforcing: a shorter equation is more likely to be useful to

shorten others,

I large systems: # pairings of n equations grows as O(n2)

I sparse systems will reveal a better elimination strategy,I PDEs more likely to become total derivatives or ODEs,I random polynomials with fewer terms factorize much more

likely than large polynomials.

An algorithm and implementation are described in J. Symb.Comp. 33, no 3 (2002).

shorten others,I large systems: # pairings of n equations grows as O(n2)

I sparse systems will reveal a better elimination strategy,

I PDEs more likely to become total derivatives or ODEs,I random polynomials with fewer terms factorize much more

I sparse systems will reveal a better elimination strategy,I PDEs more likely to become total derivatives or ODEs,

I random polynomials with fewer terms factorize much morelikely than large polynomials.

Handling Nonlinearity

I different heuristics for splitting a computation intosub-cases by:

I taking the most frequent factor to be = 0 or 6= 0,I taking the most frequent unknown to be = 0 or 6= 0,I taking expressions to be = 0 or 6= 0 which are either

specified interactively, or given in a specific order at thestart of the computation,

I a heuristic that picks one of the factorizable equations andan order of their factors to be set to zero,

I initiating case splittings when the system becomes criticallylarge

Inequations

I the treatment of inequations: their usage, active collectionand derivation, and their constant update in an ongoingreduction process based on newly derived equations

For example: