Computing symmetries and recursion operators of evolutionary super-systems using the SsTools environment
CComputing symmetries and recursionoperators of evolutionary super-systems usingthe SsTools environment
Arthemy V. Kiselev π and Andrey O. Krutov π and Thomas Wolf ππ Johann Bernoulli Institute for Mathematics and Computer Science, Universityof Groningen, P.O.Box 407, 9700 AK Groningen, The Netherlands. π Independent University of Moscow, Bolshoj Vlasyevskij Pereulok 11, 119002,Moscow, Russia. π Department of Mathematics and Statistics, Brock University, 500 GlenridgeAvenue, St.Catharines, Ontario, Canada L2S 3A1
Abstract
At a very informal but practically convenient level, we discuss the step-by-step computation of nonlocal recursions for symmetry algebras of nonlinearcoupled boson-fermion π = 1 supersymmetric systems by using the SsTools environment.
The principle of symmetry plays an important role in modern mathematicalphysics. The differential equations that constitute integrable models practically al-ways admit symmetry transformations. The presence of symmetry transformationin a system yields two types of explicit solutions: those which are invariant undera transformation (sub)group and the solutions obtained by propagating a know so-lution by the same group. The recursion operator is a (pseudo)differential operatorwhich maps symmetries of a given system to into symmetries of the same system.The recursion operators allow to obtain new symmetries for a given seed symmetry.It is common for important equations of mathematical physics not to have localrecursion operators other than the identity id : π β¦β π . Instead, they often admitnonlocal recursions which involve integrations such as taking the inverse of the totalderivative π· π₯ with respect to the independent variable π₯ . To describe such nonlocalstructures we use the approach of nonlocalities. By nonlocalities we mean an ex-tension of the initial system by new fields such that the initial fields are differentialconsequences of the new ones. In the case of recursion operators such fields oftenarise for conservation laws. We refer to a recursion operator for the Kortewegβde Vries equations as a motivating example of a nonlocal recursion operator, seeExample 7 on page 11.The supersymmetric integrable systems, i.e. systems involving commuting (boso-nic, or even) and anticommuting (fermionic, or odd) independent variables and/orunknown functions, have found remarkable applications in modern mathematicalphysics (for example supergravity models, perturbed conformal field theory [12]; werefer to [1, 4] for a general overview). When dealing with supersymmetric modelsof theoretical physics, it is often hard to predict whether a certain mathematical a r X i v : . [ n li n . S I] M a y A. V. Kiselev, A. O. Krutov, T. Wolfapproximation will be truly integrable or not. Therefore we apply the symboliccomputation to exhibit necessary integrability features. In what follows we restrictourselves to the case of π = 1 (where π refers to the number of odd anticommutingindependent variables π π ). Nevertheless, the techniques and computer programs de-scribed below could be easily applied to the case of arbitrary π . Usually, the π isnot bigger than 8, see for example [11] and [3]. It is an interesting open problem toestablish criteria that set a limit on π in β π -extendedβ supersymmetric equationsof mathematical physics.The latest version of SsTools can be found at [16], see also [8]. We referto [1, 4] and [2, 5, 6, 9, 13] for reviews of the geometry and supergeometry of partialdifferential equations. We refer to [10] for an overview of other software that couldbe used for similar computational tasks.
We fix notation first. Let π₯ be the independent variable, π’ = ( π’ , . . . , π’ π ) denote theunknown functions irrespective of their (anti-)commutation properties, and π’ π = { π’ ππ | π’ ππ = π π π’ π /ππ₯ π } denote the partial derivatives of π’ π of order π β N . We extendthe independent variable π₯ by the pair ( π₯, π ), where π is the Grassmann variablesuch that π = 0. The superderivative is defined as π = π π + ππ π₯ . Its square power is the spatial derivative, π = π π₯ .Fields π’ ( π₯, π‘ ) now become π = 1 superfields π’ ( π₯, π‘ ; π ). Provided that π = 0,they have a very simple Taylor expansions in π : π’ ( π₯, π‘ ; π ) = π’ ( π₯, π‘ ) + ππ’ ( π₯, π‘ ) , here π’ has the same parity as π’ and π’ has the opposite parity. The bosonic fields(those commuting with everything) are denoted by π ( π₯, π‘ ; π ), and the fermionic fields,which anti-commute between themselves and with π , will be denoted by π ( π₯, π‘ ; π ).Further, we write π’ π/ = π π π’ for the π th order super-derivative of π’ . Note that thesuper-derivatives π π +1 π are fermionic and π π +1 π are bosonic for any π β N .Computer input will be shown in text font, for example, f(i) for π π , b(j) for π π , df(b(j),x) for the derivative of π π with respect to π₯ , d(1,f(i)) for the su-perderivative π π π π written as π π π , because we will have only one π and one π .Let π, π < β be fixed integers. Suppose that πΉ πΌ ( π₯, π‘, π’, π ( π’ ) , . . . , π’ π , π’ π‘ ) is asmooth function for any integer πΌ β€ π . In what follows, we consider systems ofdifferential equations, { πΉ πΌ ( π₯, π’, π ( π’ ) , π’ , π ( π’ ) , . . . , π’ π , π’ π‘ ) = 0 } , πΌ = 1 , . . . , π, (1)of order π and, especially, the autonomous translation-invariant evolutionary sys-tems, β° = { πΉ πΌ = π’ πΌπ‘ β Ξ¦ πΌ ( π’, . . . , π’ π ) = 0 } , πΌ = 1 , . . . , π, omputing symmetries and recursion operators using SsTools 3which are resolved w.r.t. the time derivatives, and the systems obtained by extendingthe evolutionary systems with some further differential relations upon π’ βs and their(super-)derivatives. The weight technique
Physically meaningful equations have often several symmetries, among them one ormore scaling symmetries. Suppose that to each super-field π’ π and to the derivative π π‘ one can assign a real number (the weight), which is denoted by [ π΄ ] for any object π΄ .By definition, [ π π₯ ] β‘
1. The weight of a product of two objects is the sum of theweights of the factors, whence [ π ] = . The weight of any nonzero constant equals0, but the weight of a zero-valued constant can be arbitrary.From now on, we consider differential equations (1) with differential-polynomials πΉ πΌ that admit the introduction of weights for all variables and derivations such thatthe weights of all monomials in each equation πΉ πΌ = 0 coincide. These equationsare scaling-invariant, or homogeneous. Example 1.
Consider the Burgers equation π π‘ = π π₯π₯ β ππ π₯ , π = π ( π₯, π‘ ) . (2)The weights are uniquely defined,[ π ] = 1 , [ π π‘ ] = 2 β [ π‘ ] = β , [ π π₯ ] β‘ . Indeed, equation (2) is homogeneous w.r.t. these weights,[ π π‘ ] = 1 + 2 = [ π π₯π₯ ] = 1 + 1 + 1 = [2 ππ π₯ ] = 0 + 1 + 1 + 1 = 3and, clearly, this is the only way to choose the weights.A system of differential equations could be homogeneous w.r.t. to different weightsystems. For a given system of differential equations these weight systems can befound by using the FindSSWeight function from
SsTools : FindSSWeights(N,nf,nb,exli,zerowei,verbose) where N . . . the number of superfields π π ; nf . . . number of fermion fields f(1), f(2), ..., f(nf) ; nb . . . number of boson fields b(1), b(2), ..., b(nb) ; exli . . . list of equations or expressions; zerowei . . . list of constants or other kernels that should have zero weight; verbose . . . ( =t (true)/ nil (false)) whether detailed comments shall be made.The program returns a list of homogeneities, each homogeneity being a list fh,bh, hi of fh . . . a list of the weights of f(1),b(2), ..., f(nf) ; A. V. Kiselev, A. O. Krutov, T. Wolf bh . . . a list of the weights of b(1),b(2), ..., b(nb) ; hi . . . a list of the weights of equations/expressions in the input.Weights are scaled such that weight of [ π π₯ ] is 2, i.e. the weight of any π is 1. Sothe computer weights will be twice the βusualβ weight. Input expressions can be infield form or coordinate form. Example 2.
Consider the nonlinear SchrΒ¨odinger equation π π‘ = π π₯π₯ + 2( π ) π , π = π ( π₯, π‘ ) ,π π‘ = β π π₯π₯ β π ) π , π = π ( π₯, π‘ ) . We compute all possible weight systems of this system of equations:
FindSSWeights(0,0,2,{df(b(1),t) = df(b(1),x,2) + 2*b(1)**2*b(2),df(b(2),t) = - df(b(2),x,2) - 2*b(2)**2*b(1) },{},t)$
The output contains
This system has the following homogeneities:W[t] = -4W[b(2)] = - arbcomplex(1) + 4W[b(1)] = arbcomplex(1)W[x] = -2 which gives us the following family of weight systems for (3)[ π π‘ ] = 2 , [ π ] = β π + 4 , [ π ] = π, [ π π₯ ] = 1 , where π is an arbitrary constant. Definition 1. An π th order symmetry of an evolutionary system β° = { π’ π‘ = Ξ¦ } isanother autonomous evolutionary system β° β² = { π’ π = π ( π₯, π’, π ( π’ ) , . . . , π’ π ) } upon π’ ( π , π‘, π₯ ; π ) such that a solution of the Cauchy problem for β° β² propagates solutionsof β° to solutions of β° . A necessary and sufficient condition for a vector π to be asymmetry of β° is that solutions of the system satisfy π· π‘ π· π π’ = Β± π· π π· π‘ π’ (4)where the minus sign applies if both π‘ and π are fermionic. Because (4) is to besatisfied by solutions π’ of the system β° , i.e. π’ π‘ is replaced by Ξ¦ giving the so-calledlinearization Lin β° of the system β° :Lin Ξ¦ ( π ) := π· π‘ π = Β± π· π Ξ¦ . This is a linear system for π . If π is a bosonic variable then the symmetry isa system β° β² = { π π = πΉ, π π = π΅ } and if π is fermionic then the symmetry is β° β² = { π π = π΅, π π = πΉ } .omputing symmetries and recursion operators using SsTools 5The understanding of linearized systems Lin β° from a computational viewpoint isas follows; we consider the differential polynomial case since this is what SsTools can be applied to. Let us first consider the case of bosonic π .Given a system β° of super-equations, formally assign the new βlinearizedβ fields πΉ π = f(nf+i) and π΅ π = b(nb+j) to π π = f(i) and π π = b(j) , respectively, with π = 1, . . . , nf and π = 1, . . . , nb . Pass through all equations, and whenever a powerof a derivative of a variable π π or π π is met, differentiate (in the usual sense) thispower with respect to its base, multiply the result from the right by the same orderderivative of πΉ π or π΅ π , respectively, and insert the product in the position wherethe power of the derivative was met. Now proceed by the Leibniz rule. The finalresult, when all equations in the system β° are processed, is the linearized systemLin β° .If π is fermionic then proceed in the same way, except we get extra factors of β β An overall factor β π‘ is fermionic due to anticommuting π· π‘ and π· π . β When differentiating a factor in a product then a factor β β When changing the order of π· π and π in differentiating then a factor of β π is the number of newβlinearizedβ fields πΉ π , π΅ π that are introduced in the linearized equation. For bosonic π these are πΉ = f(nf+1) . . . f(nf+nf) , π΅ = b(nb+1) . . . b(nb+nb) whereas forfermionic π these are πΉ = f(nf+1) . . . f(nf+nb) , π΅ = b(nb+1) . . . b(nb+nf) .To summarize, the linearization is obtained by a complete differentiation π· π Ξ¦applying the Leibniz rule and chain rule (in place) and substituting π’ π = π . Example 3.
The linearized counterpart of π π‘ = π ( π π ) is π΅ π‘ = π΅ ( π π ) + 2 π π π π πΉ for bosonic symmetry parameter π and πΉ π‘ = πΉ ( π π ) β π π π π π΅ for fermionicΒ― π . Likewise, for π π‘ = ππ π π₯ ( π·π ) and parity-odd Β― π , the linearization is πΉ π‘ = πΉ π π π₯ ( π·π ) + ππ΅π π₯ ( π·π ) β ππ π΅ π₯ ( π·π ) β ππ π π₯ π·π π·π΅ .The scaling weights of the new fields are always set by [ πΉ ] = [ π ], [ π΅ ] = [ π ] forbosonic π and [ πΉ ] = [ π ], [ π΅ ] = [ π ] for fermionic Β― π .The linearization Lin β° for a system of evolution equations β° is obtained usingthe procedure linearize : linearize(pdes, nf, nb, tpar, spar); where pdes . . . list of equations π ππ‘ = Ξ¦ ππ , π ππ‘ = Ξ¦ ππ ; nf . . . number of the fermion fields f(1), f(2), ..., f(nf) ; nb . . . number of the boson fields b(1), b(2), ..., b(nb) ; tpar . . . (= t (true)/ nil (false)) whether π‘ is parity changing or not spar . . . (= t (true)/ nil (false)) whether π is parity changing or not A. V. Kiselev, A. O. Krutov, T. Wolf Example 4.
The linearizations of the system with parity reversing time Β― π‘ , π Β― π‘ = π + π π, π Β― π‘ = π π + π π, (5)are obtained as follows, depend {b(1),f(1)},x,t;linearize({df(f(1),t)=b(1)**2+d(1,f(1)),df(b(1),t)=f(1)*b(1)+d(1,b(1))},1,1,t,nil); for bosonic π , and the same call with t as last parameter instead of nil for fermionicΒ― π . The result is the new system involving twice as many variables as the originalequation. The linearization correspondence between the fields is π β¦β πΉ ( f(1) β¦β f(2) ) , π β¦β π΅ ( b(1) β¦β b(2) )for bosonic π and π β¦β π΅ ( f(1) β¦β b(2) ) , π β¦β πΉ ( b(1) β¦β f(2) )for fermionic Β― π .The procedure to compute the linearization is the same for normal times π‘ andfor parity reversing times Β― π‘ except of a factor (-1) of the rhsβs π΅, πΉ if both Β― π‘, Β― π arefermionic (because of the anticommutativity of π· Β― π‘ , π· Β― π in that case).The linearized system incorporates β the initial system: df(f(1),t) = b(1)**2 + d(1,f(1)),df(b(1),t) = f(1)*b(1) + d(1,b(1)), β and its linearizations: df(f(2),t) = 2*b(2)*b(1) + d(1,f(2)),df(b(2),t) = d(1,b(2)) + f(2)*b(1) + f(1)*b(2) for π ; respectively, df(b(2),t) = d(1,b(2)) - 2*f(2)*b(1),df(f(2),t) = - b(2)*b(1) + d(1,f(2)) - f(2)*f(1) for Β― π .One does not need to compute the linearizations in order to obtain a symmetryof a differential equation. However, the explicit computation of the linearizationswill be required for finding the recursions, which are βsymmetries of symmetries.βFor computing symmetries of any system β° , use the procedure ssym with the call ssym(N,tw,sw,afwlist,abwlist,eqnlist,fl,inelist,flags);N . . . the number of superfields π π ; tw . . . Γ the weight of π π‘ ; sw . . . Γ the weight of π π ;omputing symmetries and recursion operators using SsTools 7 afwlist . . . list of 2 Γ weights of the fermion fields f(1),f(2), . . . ,f(nf) ; abwlist . . . list of 2 Γ weights of the boson fields b(1),b(2), . . . ,b(nb) ; eqnlist . . . list of extra conditions on the undetermined coefficients; fl . . . extra unknowns in eqnlist to be determined; inelist . . . a list, each element of it is a non-zero expression or a list with at leastone of its elements being non-zero; flags . . . list of flags: init : only initialization of global data, zerocoeff : all coefficients = 0 which do not appear in inelist, tpar : if the time variable π‘ changes parity, spar : if the symmetry variable π changes parity, lin : if symmetries of a linearization are to be computed, filter : if a symmetry should satisfy homogeneity weights defined in hom wei Note that the computer representation of the weights is twice the standard notation;thus we avoid half-integers values for convenience. For more details on other flagsrun the command sshelp() . Definition 2.
A recursion operator for the symmetry algebra of an evolutionarysystem β° = { π ππ‘ = Ξ¦ ππ , π ππ‘ = Ξ¦ ππ } , π = 1 , . . . , π π , π = 1 , . . . , π π is the vector expression β = ( β π ( π ) , . . . , β π π π ( π ) , β π ( π ) , . . . , β π π π ( π )) π , which is linear w.r.t. the new fields π , which for bosonic π are π = ( πΉ , . . . , πΉ π π , π΅ ,. . . , π΅ π π ) π and for fermionic Β― π are π = ( π΅ , . . . , π΅ π π , πΉ , . . . , πΉ π π ) π , and their deriva-tives and which is a (right-hand side of a) symmetry of β° whenever π = ( πΉ , . . . , πΉ π π , π΅ , . . . , π΅ π π ) π for π , or π = ( π΅ , . . . , π΅ π π , πΉ , . . . , πΉ π π ) π for Β― π is a symmetry of β° . In other words, β : sym β° β sym β° is a linear operator thatgenerates a symmetry of β° when applied to a symmetry π β sym β° .The weight [ β ] of recursion β is the difference [ β ( π )] β [ π ] of the weights of the(time derivatives π π β² , π π of π’ , i.e. of the) resulting and the initial symmetries, here π β sym β° .Different recursions can have the same weight. Experiments show that in thiscase operators may have different properties, e.g., a majority of them is nilpotent,several zero-order operators act through multiplication by a differential-functional A. V. Kiselev, A. O. Krutov, T. Wolfexpression and do not increase the differential orders of the flows, and only fewrecursions construct higher-order symmetries and reveal the integrability.If [ β ] is the weight of a recursion β , then, clearly, at least one recursion isfound with weight 2 Γ [ β ]. Indeed, this is β . At the same time, other recursionsmay appear with weight N Γ [ β ]. If β is nonlocal (see below), then its powersare also nonlocal, but it remains a very delicate matter to predict the form of theirnonlocalities, and strong theoretical assertions can be formulated for some particularintegrable system. Remark 1.
The derivation of the weight of a recursion is constructive in the follow-ing sense. To attempt finding a recursion for an evolutionary system, it is beneficialto know already many symmetries π π π of different weights [ π π ]. Then one tries firstthe weights [ β ] := [ π π ] β [ π π ] for various π, π . However, the recursions obtained thisway can be nilpotent, i.e. β π ( π ) β‘ π and any π . More promising areweights for which there exist π, π, π with [ β ] = [ π π ] β [ π π ] = [ π π ] β [ π π ] and π π , π π , π π being elements of the infinite hierarchies of symmetries with low weights. Still, theactual weights of unknown recursions can turn out to be larger than the weightdifferences of the lowest order symmetries.Finally, non-trivial recursions may only appear in nonlocal settings. We discussthis in section 4.The crucial point is that β is a symmetry of the linearized system Lin β° . Theoriginal system β° is only used for substitutions. Hence we use ssym for findingrecursions of the linearizations, which are previously calculated by linearize . Example 5.
Let us construct a recursion for equation (2). We obtain the lineariza-tion using the procedure linearize , linearize({df(b(1),t)=df(b(1),x,2)-2*b(1)*df(b(1),x)},0,1,nil,nil); The new system depends on the two fields π = b(1) and π΅ = b(2) . df(b(2),t) = -2*b(2)*df(b(1),x)-2*b(1)*df(b(2),x)+df(b(2),x,2),df(b(1),t) = df(b(1),x,2)-2*b(1)*df(b(1),x). The recursion of weight 0 is obtained as follows (thus sw = 2[ β ] = 0). ssym(1, 4, 0, {}, {2, 2},{df(b(2),t)= -2*b(2)*df(b(1),x)-2*b(1)*df(b(2),x)+df(b(2),x,2),df(b(1),t)=> df(b(1),x,2)-2*b(1)*df(b(1),x)}, {}, {}, {lin}); By writing df(b(1),t)=> ... we require that the original system is only used forsubstitutions.The output contains df(b(2),s)=b(2)1 solution was found. omputing symmetries and recursion operators using SsTools 9This is the identity transformation π β¦β β = id ( π ) β‘ π ;it maps symmetries to themselves. Clearly, the identity is a recursion for anysystem ! Example 6 (A recursion for the Kortewegβde Vries equation) . The KdV equationupon the bosonic field π’ ( π₯, π‘ ) is π’ π‘ = β π’ π₯π₯π₯ + π’π’ π₯ . (6)Equation (6) is homogeneous w.r.t. the weights[ π’ ] = 2 , [ π‘ ] = β , [ π₯ ] = β . The linearization of (6) is constructed using the procedure linearize , linearize({df(b(1),t)= -df(b(1),x,3) + b(1)*df(b(1),x)},0,1,nil,nil) Thus we obtain the new system that depends on the fields π’ = b(1) , π = b(2) : df(b(1),t) = - df(b(1),x,3) + b(1)*df(b(1),x)df(b(2),t) = b(2)*df(b(1),x) + b(1)*df(b(2),x) - df(b(2),x,3). As a first guess we are looking for the recursion operator of weight 0. The recursionof weight 0 (hence sw = β β ] = 0) is constructed as follows. ssym(1, 6, 0, {}, {4, 4},{df(b(2),t) = b(2)*df(b(1),x)+b(1)*df(b(2),x)-df(b(2),x,3),df(b(1),t) => -df(b(1),x,3)+b(1)*df(b(1),x)},{},{},{lin}); The output contains df(b(2),s)=b(2)1 solution was found.
Again, this operator β of weight 0 is the identity, π β¦β β ( π ) = (id) ( π ) . The well-known explanation for this result is that, as a rule, one needs to introducenonlocalities first and only then obtains nontrivial recursions in the nonlocal setting.0 A. V. Kiselev, A. O. Krutov, T. Wolf
The nonlocal variables for π = 1 super-systems are constructed by trivializing [7, 9]conservation laws π π‘ (density) . = π (super-flux) , that is, in each case the above equality holds by virtue ( . =) of the system at hand andall possible differential consequences from it. The standard procedure [9] suggeststhat every conserved current determines the new nonlocal variable, say π£ , whosederivatives are set to π£ π‘ = super-flux , π π£ = density (7a)if the time π‘ preserves the parities and π£ Β― π‘ = β super-flux , π π£ = density (7b)if the time Β― π‘ is parity-reversing. Note that in the classical case the nonlocality π£ canbe specified through π£ π‘ = flux , π£ π₯ = density (7c)for the conservation law π π‘ (density) . = π π₯ (flux). Each nonlocality thus makes theconserved current trivial because the cross derivatives of π£ coincide in this case,[ π· π‘ , π ]( π£ ) = 0, where [ , ] stands for the commutator if π‘ is parity-preserving andfor the anti commutator whenever the time Β― π‘ is parity-reversing. The new variablescan be bosonic or fermionic; the parities are immediately clear from the formulaefor their derivatives.Hence, starting with an equation β° , one calculates several conserved currentsfor it and trivializes them by introducing a layer of nonlocalities whose derivativesare still local differential functions. This way the number of fields is increased andthe system is extended by new substitution rules. Moreover, it may acquire newconserved currents that depend on the nonlocalities and thus specify the second layerof nonlocal variables with nonlocal derivatives. At each step the number of variableswill increases by 2 compared with the previous layer (a new nonlocal variable plusthe corresponding linearized field). Clearly, the procedure is self-reproducing.So, one keeps computing conserved currents and adding the layers of nonlinear-ities until an extended system Λ β° is achieved such that its linearization Lin Λ β° has asymmetry β ; this symmetry of Lin Λ β° is a recursion for the extended system Λ β° .The calculation of conservation laws for evolutionary super-systems with homo-geneous polynomial right-hand sides is performed by using the procedure ssconl : ssconl(N,tw,mincw,maxcw,afwlist,abwlist,pdes); where N . . . the number of superfields π π ;omputing symmetries and recursion operators using SsTools 11 tw . . . 2 Γ the weight [ π π‘ ]; mincw . . . minimal weight of the conservation law; maxcw . . . maximal weight of the conservation law; afwlist . . . list of weights of the fermionic fields f(1) , . . . , f(nf) ; abwlist . . . list of weights of the bosonic fields b(1) , . . . , b(nb) ; pdes . . . list of the equations for which a conservation law must be found.For positive weights of bosonic variables, the ansatz is fully determined throughthe weight mincw, ..., maxcw of the conservation law. If a boson weight is non-positive then the global variable max_deg must have a positive integer value whichis the highest degree of such a variable or any of its derivatives in any ansatz.The conservation law condition leads to an algebraic system for the undeterminedcoefficients, which is further solved automatically by Crack .Having obtained a conserved current, one defines the new bosonic or fermionicdependent variable (the nonlocality) using the standard rules (7).We illustrate the general scheme of fixing the derivatives of a nonlocal variable byseveral examples. Further information on the
SsTools environment is containedin [8] and the sshelp() function in
SsTools . The algebraic structures that describethe geometry of recursion operators for super-PDE are described in detail in [9].Some more examples and their applications are also found in [7].
Example 7 (A nonlocal recursion for the KdV equation) . Consider the Kortewegβde Vries equation (6) again, π π‘ ( π’ ) . = π π₯ (οΈ β π’ π₯π₯ + 12 π’ )οΈ . We declare that the conserved density π’ is the spatial derivative π€ π₯ = π’ of a newnonlinear variable π€ and the flux is its derivative w.r.t. the time, π€ π‘ = β π’ π₯π₯ + π’ .Then π€ π₯π‘ = π€ π‘π₯ by virtue of (6). Thus we introduce the bosonic nonlocality π€ bytrivializing the conserved current. Let us remember that π€ π₯ = π’,π€ π‘ = β π€ π₯π₯π₯ + 12 π€ π₯ and the weight of π€ is [ π€ ] = 1 because [ π€ ] + [ π π₯ ] = [ π’ ] = 2.Next, we compute the linearization of equation (6) and of the relations thatspecify the new variable, linearize({df(b(1),t)= -df(b(1),x,3) + b(1)*df(b(1),x),df(b(2),x)= b(1),df(b(2),t)= -df(b(2),x,3) + df(b(2),x)**2}, 0, 2); The linearization correspondence between the fields is π’ β¦β π ( b(1) β¦β b(3) ) , π€ β¦β π ( b(2) β¦β b(4) ) . The linearized system is2 A. V. Kiselev, A. O. Krutov, T. Wolf df(b(3),t) = b(3)*df(b(1),x) + b(1)*df(b(3),x) - df(b(3),x,3),df(b(4),x) = b(3),df(b(4),t) = -df(b(4),x,3) + df(b(2),x)*df(b(4),x)
In this nonlocal setting, we obtain the nonlocal recursion of weight sw = 2[ β ] = 4as follows, ssym(1, 6, 4, {}, {4, 2, 4, 2},{df(b(3),t) = b(3)*df(b(1),x) + b(1)*df(b(3),x) - df(b(3),x,3),df(b(1),t) => b(1)*df(b(1),x) - df(b(1),x,3),df(b(2),x) => b(1),df(b(2),t) => -df(b(2),x,3) + 1/2 * df(b(2),x)**2,df(b(4),x) => b(3),df(b(4),t) => -df(b(4),x,3) + df(b(2),x)*df(b(4),x)}, {}, {}, {lin}); We recall here that only the linearized system should be written as equations, andall other relations, including the nonlocalities, should be written as substitutions.This yields the solution df(b(3),s) = -3*df(b(3),x,2) + 2*b(1)*b(3) + df(b(1),x)*b(4), which is the well-known nonlocal recursion operator for KdV, π β¦β β = (οΈ β π· π₯ + 2 π’ + π’ π₯ Β· π· β π₯ )οΈ ( π ) . (8)This recursion generates the hierarchy of local symmetries starting from the trans-lation π = π’ π₯ . The powers β , β , . . . of the recursion operator are also nonlocal. Example 8.
Consider the Burgers equation (2) and introduce the bosonic non-locality π€ of weight [ π€ ] = [ π ] β [ π π₯ ] = 0 by trivializing the conserved current π π‘ ( π ) . = π π₯ (οΈ π π₯ β π )οΈ . We therefore, set π€ π₯ = π, π€ π‘ = π€ π₯π₯ β π€ π₯ . (9)The linearization of the extended system is obtained through linearize({df(b(1),t)=df(b(1),x,2) - 2*b(1)*df(b(1),x),df(b(2),x)=b(1),df(b(2),t)=df(b(2),x,2) - df(b(2),x)**2}, 0, 2); The correspondence between the bosonic fields is π β¦β π΅ ( b(1) β¦β b(3) ) , π€ β¦β π ( b(2) β¦β b(4) ) . The entire linearized system is (2) and (9) together with the relations df(b(3),t)= -2*b(3)*df(b(1),x)-2*b(1)*df(b(3),x)+df(b(3),x,2);df(b(4),x)= b(3);df(b(4),t)= df(b(4),x,2) - 2*df(b(4),x)*df(b(2),x). omputing symmetries and recursion operators using SsTools 13The difference between weights of the first-order and the second-order symmetriesis 1. Hence, the recursion operator could have weight 1, see Remark 1. The nonlocalrecursion of weight 1, sw = 2[ β ] = 2, is obtained by max_deg:=1;ssym(1, 4, 2, {}, {2, 0, 2, 0},{df(b(3),t) = -2*b(3)*df(b(1),x)-2*b(1)*df(b(3),x)+df(b(3),x,2),df(b(4),x) => b(3),df(b(4),t) => df(b(4),x,2)-2*df(b(4),x)*df(b(2),x),df(b(1),t) => -2*b(1)*df(b(1),x)+df(b(1),x,2),df(b(2),x) => b(1),df(b(2),t) => df(b(2),x,2)-df(b(2),x)**2}, {},{},{lin}); We finally get the recursion df(b(3),s) = -df(b(3),x) + b(1)*b(3) + df(b(1),x)*b(4), which is nonlocal, π β¦β β = (οΈ β π· π₯ + π + π π₯ π· β π₯ )οΈ ( π ) . Example 9.
Consider the super-field representation [7] of the Burgers equation,see (2), π π‘ = π π, π π‘ = π π + π ;its weights are | π | = | π | = , | π π₯ | = 1, and | π π‘ | = .We introduce the nonlocal bosonic field π€ ( π₯, π‘ ; π ) of weight [ π€ ] = [ π ] β [ π ] =1 β π π€ = β π, π€ π‘ = β π. We get the linearized system by linearize({df(f(1),t)= d(1, b(1)),df(b(1),t)= df(f(1),x) + b(1)**2,d(1,b(2)) = -f(1),df(b(2),t)= -b(1)}, 1, 2);
For the linearization correspondence between the fields is π β πΉ , π β π΅ , π€ β π ,and we have πΉ π‘ = π π΅, π΅ π‘ = 2 ππ΅ + π πΉ, π π = β πΉ, π π‘ = β π΅, that is, df(f(2),t)= d(1, b(3)),df(b(3),t)= 2*b(1)*b(3)+d(1,f(2)),d(1,b(4)) = -f(2),df(b(4),t)= -b(3). , sw = 2[ β ] = 1: theinput is max_deg:=1;ssym(1, 1, 1, {1, 1}, {1, 0, 1, 0},{df(f(2),t) = d(1, b(3)),df(b(3),t) = 2*b(3)*b(1) + d(1, f(2)),d(1, b(4)) => - f(2),df(b(4),t) => - b(3),df(f(1),t) => d(1, b(1)),df(b(1),t) => d(1, f(1)) + b(1)**2,d(1, b(2)) => -f(1),df(b(2),t) => -b(1)}, {}, {}, {lin}); The recursion is df(f(2),s)=d(1,b(3)) + d(1,b(1))*b(4) - f(2)*b(1),df(b(3),s)=b(4)*b(1)**2 + b(3)*b(1) + d(1,f(2)) + d(1,f(1))*b(4), in other words, (οΈ πΉπ΅ )οΈ β¦β β (οΈ πΉπ΅ )οΈ = (οΈ π π΅ β π π π β πΉ β πΉ π β π π β πΉ + π΅π + π πΉ β π π π β πΉ )οΈ . Example 10.
Consider the fifth order evolution superequation found by Tian andLiu (Case F in [14], see also [6, 15]): π π‘ = π π₯ + 10( π π₯π₯ π π ) π₯ + 5( π π₯ π π π₯ ) π₯ + 15 π π₯ ( π π ) + 15 π ( π π π₯ )( π π ) . (10)In what follows, we are considering this equations in components. Substitution π + ππ’ for π in (10), for example, using SsTools , we obtain π’ π‘ = π’ π₯ + 10 π’π’ π₯π₯π₯ + 20 π’ π₯ π’ π₯π₯ + 30 π’ π’ π₯ β π π₯π₯π₯ π π₯ + 15 π’π π₯π₯ π + 15 π’ π₯ π π₯ π, (11a) π π‘ = π π₯ + 10 π’π π₯π₯π₯ + 15 π’ π₯ π π₯π₯ + 5 π’ π₯π₯ π π₯ + 15 π’ π π₯ + 15 π’π’ π₯ π, (11b)where π’ is a bosonic field and π is a fermionic field.Observe that the bosonic limit ( π := 0) is the fifth order symmetry of Kortewegβde Vries equation (6). However, a direct computation shows that the equation (11)has local symmetries of the orders 1+6 π and 5+6 π , where π β N , and does not haveany local symmetries of order 3+6 π , where π β N . Therefore, the recursion operatorfor (11) should be at least of order 6. Let us also assume that the bosonic limit ofthe recursion operator for (11) is the third power β of the recursion operator (8)for the Kortewegβde Vries equation.It is easy to check that for the construction of the 3rd power of the recursion oper-ator (8) we should βtrivialiseβ the following conserved densities of the KortewegβdeVries equation: π’ , π’ + π’ π₯π₯ and π’ π₯ + 6 π’π’ π₯π₯ + 5 π’ π₯ + 2 π’ .omputing symmetries and recursion operators using SsTools 15Let π and π satisfy the linearized equation for (11). The correspondence betweenfields is the following π’ β¦β π , π β¦β π . The linearized system of nonlocalities for thegeneralisation of those conservation laws for the supersymmetric equation (11) isthe following:1) the layer of nonlocalities corresponding to the generalisation of the conserveddensity π’ of the Kortewegβde Vries equation π π₯ = π,π π‘ = β ππ’π π₯ + π (30 π’ + 10 π’ π₯π₯ β ππ π₯ ) β π π₯π₯ π π₯ + π π₯ (5 π π₯π₯ + 15 ππ’ ) + π π₯ + 10 π π₯π₯ π’ + 10 π π₯ π’ π₯ ,
2) the layer of nonlocalities corresponding to the generalisation of the conserveddensity π’ + π’ π₯π₯ of the Kortewegβde Vries equation π π₯ = 2 ππ’ + 2 π π₯ π,π π₯ = ππ’ + ππ,π π‘ = π ( β π’ π π₯ + 30 ππ’π’ π₯ ) + π (60 π’ + 40 π’π’ π₯π₯ + 2 π’ π₯ β π π₯π₯ π π₯ β ππ’π π₯ β ππ π₯π₯π₯ ) + 2 π π₯ π β π π₯ π π₯ + π π₯π₯π₯ (2 π π₯π₯ + 20 ππ’ ) + π π₯π₯ ( β π’π π₯ β π π₯π₯π₯ + 30 ππ’ π₯ ) + π π₯ (30 π’π π₯π₯ + 2 π π₯ + 60 ππ’ + 10 ππ’ π₯π₯ ) + 2 π π₯ π’ β π π₯π₯π₯ π’ π₯ + π π₯π₯ (20 π’ + 2 π’ π₯π₯ β ππ π₯ ) + π π₯ ( β π’ π₯π₯π₯ β ππ π₯π₯ ) ,π π‘ = π (15 π’ + 10 π’π’ π₯π₯ + π’ π₯ + 5 π’ π₯ β π π₯π₯ π π₯ ) + π (20 π’π π₯π₯ + π π₯ β π π₯ π’ π₯ + 45 ππ’ + 10 ππ’ π₯π₯ ) + π π₯ π’ β π π₯π₯π₯ π’ π₯ + π π₯π₯ (10 π’ + π’ π₯π₯ + 5 ππ π₯ ) + π π₯ ( β π’π’ π₯ β π’ π₯π₯π₯ β ππ π₯π₯ ) + π π₯ π β π π₯π₯π₯ π π₯ + π π₯π₯ ( π π₯π₯ + 10 ππ’ ) + π π₯ ( β π’π π₯ β π π₯π₯π₯ + 10 ππ’ π₯ ) ,
3) the layer of nonlocalites corresponding to the generalisation of the conserveddensities π’ π₯ + 6 π’π’ π₯π₯ + 5 π’ π₯ + 2 π’ of the Kortewegβde Vries equation π π₯ = β π ππ’ + 3 ππ’π π₯ + π (6 π’ + 3 ππ π₯ ) β π π₯π₯π₯ π β π π₯ ππ’ + 2 π π₯π₯ π’,π π₯ = π (6 π’ β ππ π₯ ) β π ππ’ + 14 ππ’π π₯ + 2 π π₯π₯π₯ π’ + 7 π π₯ π’ β π π₯π₯π₯ π,π π‘ = π ( β π’ π π₯π₯ β π’π π₯ + 45 π’π π₯ π’ π₯ + 9 π π₯π₯π₯ π’ π₯ β π π₯π₯ π’ π₯π₯ + 9 π π₯ π’ π₯π₯π₯ β ππ’ β ππ’π’ π₯π₯ β ππ’ π₯ β ππ’ π₯ + 45 ππ π₯π₯ π π₯ ) + π (45 π’ π π₯ + 9 π’ π π₯π₯π₯ + 3 π’π π₯ β π’π π₯π₯ π’ π₯ + 21 π’π π₯ π’ π₯π₯ β π π₯ π’ π₯ + 3 π π₯π₯π₯ π’ π₯π₯ β π π₯π₯ π’ π₯π₯π₯ + 3 π π₯ π’ π₯ + 42 π π₯ π’ π₯ β ππ’π’ π₯π₯π₯ β ππ’ π₯π₯ π’ π₯ ) + π (180 π’ + 300 π’ π’ π₯π₯ + 32 π’π’ π₯ + 138 π’π π₯π₯ π π₯ β π’ π₯π₯π₯ π’ π₯ + 16 π’ π₯π₯ + 16 π π₯ π π₯ β π π₯π₯π₯ π π₯π₯ + 135 ππ’ π π₯ + 78 ππ’π π₯π₯π₯ + 13 ππ π₯ + 93 ππ π₯π₯ π’ π₯ + 111 ππ π₯ π’ π₯π₯ ) + π π₯ π π₯ + π π₯ ( β π π₯π₯ β ππ’ ) + π π₯ (6 π’π π₯ + π π₯π₯π₯ β ππ’ π₯ ) + π π₯π₯π₯ ( β π’π π₯π₯ β π π₯ + 26 π π₯ π’ π₯ β ππ’ β ππ’ π₯π₯ ) + π π₯π₯ (69 π’ π π₯ + 16 π’π π₯π₯π₯ β π π₯π₯ π’ π₯ + 17 π π₯ π’ π₯π₯ β ππ’π’ π₯ β ππ’ π₯π₯π₯ ) + π π₯ ( β π’ π π₯π₯ + 4 π’π π₯ + 45 π’π π₯ π’ π₯ β π π₯π₯π₯ π’ π₯ β π π₯π₯ π’ π₯π₯ + 5 π π₯ π’ π₯π₯π₯ β ππ’ β ππ’π’ π₯π₯ β ππ’ π₯ β ππ’ π₯ ) + 2 π π₯ π’ β π π₯ π’ π₯ + π π₯ (26 π’ + 2 π’ π₯π₯ + 8 ππ π₯ )+ π π₯π₯π₯ (28 π’π’ π₯ β π’ π₯π₯π₯ + 22 ππ π₯π₯ ) + π π₯π₯ (120 π’ + 112 π’π’ π₯π₯ + 2 π’ π₯ β π’ π₯ + 22 π π₯π₯ π π₯ + 81 ππ’π π₯ + 48 ππ π₯π₯π₯ ) + π π₯ (180 π’ π’ π₯ + 48 π’π’ π₯π₯π₯ β π’ π₯π₯ π’ π₯ + 16 π π₯π₯π₯ π π₯ + 3 ππ’π π₯π₯ + 32 ππ π₯ + 174 ππ π₯ π’ π₯ ) + ππ π₯ , π π‘ = π (90 π’ + 120 π’ π’ π₯π₯ + 12 π’π’ π₯ β π’π π₯π₯ π π₯ β π’ π₯π₯π₯ π’ π₯ + 6 π’ π₯π₯ β π π₯ π π₯ + 12 π π₯π₯π₯ π π₯π₯ β ππ’ π π₯ β ππ’π π₯π₯π₯ β ππ π₯ β ππ π₯π₯ π’ π₯ β ππ π₯ π’ π₯π₯ )+ π (45 π’ π’ π₯ + 24 π’ π’ π₯π₯π₯ + 84 π’π’ π₯π₯ π’ π₯ β π’π π₯π₯π₯ π π₯ β π’ π₯ β π π₯π₯ π π₯ π’ π₯ + 60 ππ’ π π₯π₯ β ππ’π π₯ β ππ’π π₯ π’ π₯ β ππ π₯π₯π₯ π’ π₯ β ππ π₯π₯ π’ π₯π₯ β ππ π₯ π’ π₯π₯π₯ )+ π ( β π’ π π₯π₯ β π’π π₯ + 30 π’π π₯ π’ π₯ + 6 π π₯π₯π₯ π’ π₯ β π π₯π₯ π’ π₯π₯ + 6 π π₯ π’ π₯π₯π₯ β ππ’ β ππ’π’ π₯π₯ β ππ’ π₯ β ππ’ π₯ + 30 ππ π₯π₯ π π₯ ) + π (660 π’ π π₯ + 288 π’ π π₯π₯π₯ + 34 π’π π₯ β π’π π₯π₯ π’ π₯ + 590 π’π π₯ π’ π₯π₯ β π π₯ π’ π₯ + 34 π π₯π₯π₯ π’ π₯π₯ β π π₯π₯ π’ π₯π₯π₯ + 34 π π₯ π’ π₯ + 28 π π₯ π’ π₯ + 135 ππ’ π’ π₯ β ππ’π’ π₯π₯π₯ β ππ’ π₯ β ππ’ π₯π₯ π’ π₯ β ππ π₯π₯π₯ π π₯ ) β π π₯ π’ π₯ + π π₯ (27 π’ + 2 π’ π₯π₯ β ππ π₯ ) + π π₯ (36 π’π’ π₯ β π’ π₯π₯π₯ β ππ π₯π₯ )+ π π₯π₯π₯ (106 π’ + 124 π’π’ π₯π₯ + 2 π’ π₯ β π’ π₯ β π π₯π₯ π π₯ + 72 ππ’π π₯ )+ π π₯π₯ (121 π’ π’ π₯ + 36 π’π’ π₯π₯π₯ β π’ π₯π₯ π’ π₯ + 10 π π₯π₯π₯ π π₯ β ππ’π π₯π₯ + 20 ππ π₯ + 144 ππ π₯ π’ π₯ ) + π π₯ (165 π’ + 295 π’ π’ π₯π₯ + 24 π’π’ π₯ + 28 π’π’ π₯ β π’π π₯π₯ π π₯ β π’ π₯π₯π₯ π’ π₯ + 12 π’ π₯π₯ + 10 π π₯ π π₯ β π π₯π₯π₯ π π₯π₯ β ππ’π π₯π₯π₯ + 10 ππ π₯ β ππ π₯π₯ π’ π₯ + 30 ππ π₯ π’ π₯π₯ ) + 2 π π₯ π π₯ + π π₯ ( β π π₯π₯ β ππ’ ) + π π₯ (44 π’π π₯ + 2 π π₯π₯π₯ β ππ’ π₯ ) + π π₯π₯π₯ (16 π’π π₯π₯ β π π₯ + 36 π π₯ π’ π₯ β ππ’ β ππ’ π₯π₯ )+ π π₯π₯ (325 π’ π π₯ + 104 π’π π₯π₯π₯ β π π₯π₯ π’ π₯ + 94 π π₯ π’ π₯π₯ β ππ’π’ π₯ β ππ’ π₯π₯π₯ )+ π π₯ (91 π’ π π₯π₯ + 56 π’π π₯ + 386 π’π π₯ π’ π₯ β π π₯π₯π₯ π’ π₯ β π π₯π₯ π’ π₯π₯ + 46 π π₯ π’ π₯π₯π₯ + 45 ππ’ β ππ’π’ π₯π₯ β ππ’ π₯ β ππ’ π₯ β ππ π₯π₯ π π₯ ) + 2 π’π π₯ β ππ π₯ . Here π , π , π are bosonic fields and π , π are fermionic fields. Let us notethat the generalisation of the conservation law of (6) with the density π’ π₯ + 6 π’π’ π₯π₯ +5 π’ π₯ + 2 π’ is no longer a local conservation law for (11).The weights of fields are as follows | π₯ | = β , | π‘ | = β , | π’ | = | π | = 2 , | π | = | π | = , | π | = 1 , | π | = 3 , | π | = 5 , | π | = , | π | = . Using the technique described above we obtain the following recursion operator(cf. [14]) β (οΈ(οΈ ππ )οΈ)οΈ = (οΈ π β² π β² )οΈ , where π β² = 3 π π π₯ + π ( β π’π π₯π₯ β π π₯ β π π₯ π’ π₯ β ππ’ ) + π ( β π’ π π₯ β π’π π₯π₯π₯ + 2 π π₯ β π π₯π₯ π’ π₯ β π π₯ π’ π₯π₯ + 18 ππ’π’ π₯ β ππ’ π₯π₯π₯ ) + 4 π π’ π₯ + π (24 π’π’ π₯ + 4 π’ π₯π₯π₯ + 6 ππ π₯π₯ ) + π (120 π’ π’ π₯ + 40 π’π’ π₯π₯π₯ + 4 π’ π₯ + 80 π’ π₯π₯ π’ π₯ β π π₯π₯π₯ π π₯ β ππ’π π₯π₯ β ππ π₯ π’ π₯ ) + π (128 π’ + 192 π’π’ π₯π₯ + 24 π’ π₯ + 144 π’ π₯ β π π₯π₯ π π₯ β ππ’π π₯ β ππ π₯π₯π₯ ) β π π₯ π π₯ + π π₯π₯π₯ ( β π π₯π₯ + 36 ππ’ ) + π π₯π₯ ( β π’π π₯ + 8 π π₯π₯π₯ + 58 ππ’ π₯ ) + π π₯ ( β π’π π₯π₯ + 8 π π₯ β π π₯ π’ π₯ + 72 ππ’ + 30 ππ’ π₯π₯ ) + 2 π π₯ + 24 π π₯ π’ + 60 π π₯π₯π₯ π’ π₯ + π π₯π₯ (96 π’ + 80 π’ π₯π₯ β ππ π₯ )+ π π₯ (280 π’π’ π₯ + 60 π’ π₯π₯π₯ β ππ π₯π₯ ) , omputing symmetries and recursion operators using SsTools 17 π β² = 3 π π’ + π (42 π’π’ π₯ + 6 π’ π₯π₯π₯ ) + π (35 π’ + 48 π’π’ π₯π₯ + 2 π’ π₯ + 40 π’ π₯ β π π₯π₯ π π₯ + 4 ππ π₯π₯π₯ ) + 4 π π π₯ + π ( β π’π π₯ β π π₯π₯π₯ β ππ’ π₯ ) + π (60 π’ π π₯ + 40 π’π π₯π₯π₯ + 4 π π₯ + 60 π π₯π₯ π’ π₯ + 20 π π₯ π’ π₯π₯ + 60 ππ’π’ π₯ ) + π (114 π’π π₯π₯ + 22 π π₯ + 104 π π₯ π’ π₯ + 72 ππ’ + 30 ππ’ π₯π₯ ) + 2 π π₯ + 24 π π₯ π’ + 48 π π₯π₯π₯ π’ π₯ + π π₯π₯ (54 π’ + 38 π’ π₯π₯ + 4 ππ π₯ )+ π π₯ (126 π’π’ π₯ + 14 π’ π₯π₯π₯ ) + 12 π π₯π₯π₯ π π₯ + π π₯π₯ (42 π π₯π₯ + 42 ππ’ )+ π π₯ (100 π’π π₯ + 46 π π₯π₯π₯ + 54 ππ’ π₯ ) . In [6] this system of nonlocalities was used to construct a zero-curvature repre-sentation of (11) to prove its integrability.
Acknowledgements
This work was supported in part by an NSERC grant to T. Wolf who is thanked byA. V. K. for warm hospitality. This research was done in part while A. O. K. wasvisiting at New York University Abu Dhabi; the hospitality and warm atmosphereof this institution are gratefully acknowledged.
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