Symbolic Computation of Recursion Operators for Nonlinear Differential-Difference equations
SSYMBOLIC COMPUTATION OF RECURSION OPERATORS FOR NONLINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS
Ünal Göktaş and Willy Hereman Department of Computer Engineering, Turgut Özal University Keçiören, Ankara 06010, Turkey. [email protected] Department of Mathematical and Computer Sciences, Colorado School of Mines Golden, Colorado 80401-1887, U.S.A. [email protected]
Abstract-
An algorithm for the symbolic computation of recursion operators for sys-tems of nonlinear differential-difference equations (DDEs) is presented. Recursion op-erators allow one to generate an infinite sequence of generalized symmetries. The exis-tence of a recursion operator therefore guarantees the complete integrability of the DDE. The algorithm is based in part on the concept of dilation invariance and uses our earlier algorithms for the symbolic computation of conservation laws and generalized symmetries. The algorithm has been applied to a number of well-known DDEs, including the Kac-van Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which recursion opera-tors are shown. The algorithm has been implemented in
Mathematica , a leading com-puter algebra system. The package
DDERecursionOperator.m is briefly discussed.
Keywords-
Conservation Law, Generalized Symmetry, Recursion Operator, Nonlinear Differential-Difference Equation
1. INTRODUCTION
A number of interesting problems can be modeled with nonlinear differential-difference equations (DDEs) [1]-[3], including particle vibrations in lattices, currents in electrical networks, and pulses in biological chains. Nonlinear DDEs also play a role in queuing problems and discretizations in solid state and quantum physics, and arise in the numerical solution of nonlinear PDEs. The study of complete integrability of nonlinear DDEs largely parallels that of nonlinear partial differential equations (PDEs) [4]-[7]. Indeed, as in the continuous case, the existence of large numbers of generalized (higher-order) symmetries and conserved densities is a good indicator for complete integrability. Albeit useful, such predictors do not provide proof of complete integrability. Based on the first few densities and symme-tries, quite often one can explicitly construct a recursion operator which maps higher-order symmetries of the equation into new higher-order symmetries. The existence of a recursion operator, which allows one to generate an infinite set of such symmetries step-by-step, then confirms complete integrability. There is a vast body of work on the complete integrability of DDEs. Consult, e.g., [5, 8] for additional references. In this article we describe an algorithm to symboli- cally compute recursion operators for DDEs. This algorithm builds on our related work for PDEs and DDEs [9]-[11] and work by Oevel et al [12] and Zhang et al [13]. In contrast to the general symmetry approach in [5], our algorithms rely on spe-cific assumptions. For example, we use the dilation invariance of DDEs in the construc-tion of densities, higher-order symmetries, and recursion operators. At the cost of gene-rality, our algorithms can be implemented in major computer algebra systems. Our
Mathematica package
InvariantsSymmetries.m [14] computes densities and generalized symmetries, and therefore aids in automated testing of complete inte-grability of semi-discrete lattices. Our new
Mathematica package
DDERecursionOpe-rator.m [15] automates the required computations for a recursion operator. The paper is organized as follows. In Section 2, we present key definitions, ne-cessary tools, and prototypical examples, namely the Kac-van Moerbeke (KvM) [16] and Toda [17, 18] lattices. An algorithm for the computation of recursion operators is outlined in Section 3. Usage of our package is demonstrated on an example in Section 4. Section 5 covers two additional examples, namely the Ablowitz-Ladik (AL) [19] and RelativisticToda (RT) [20] lattices. Concluding remarks about the scope and limitations of the algorithm are given in Section 6.
2. KEY DEFINITIONS 2.1. Definition
A nonlinear DDE is an equation of the form ,,...),,(..., nnnn uuu Fu (1) where n u and F are vector-valued functions with N components. The subscript n cor-responds to the label of the discretized space variable; the dot denotes differentiation with respect to the continuous time variable . t Throughout the paper, for simplicity we denote the components of n u by ,...),,( nnn wvu and write ),( n uF although F typically depends on n u and a finite number of its forward and backward shifts. We assume that F is polynomial with constant coefficients. No restrictions are imposed on the shifts or the degree of nonlinearity in . F The Kac-van Moerbeke (KvM) lattice [16], also known as the Volterra lattice, ,)( nnnn uuuu (2) arises in the study of Langmuir oscillations in plasmas, population dynamics, etc. One of the earliest and most famous examples of completely integrable DDEs is the Toda lattice [17,18], ,)exp()exp( nnnnn yyyyy (3) where n y is the displacement from equilibrium of the n th particle with unit mass under an exponential decaying interaction force between nearest neighbors. With the change of variables, ),exp(, nnnnn yyvyu due to Flaschka [21], lattice (3) can be writ-ten in polynomial form [22] .)(, nnnnnnn uuvvvvu (4) A DDE is said to be dilation invariant if it is invariant under a scaling (dilation) symmetry.
Lattice (2) is invariant under ),,(),( nn utut where is an arbitrary scaling parameter. Equation (4) is invariant under the scaling symmetry ,),,(),,( nnnn vutvut (5) where is an arbitrary scaling parameter. We define the weight, , w of a variable as the exponent in the scaling parameter which multiplies the variable. As a result of this definition, t is always replaced by t and .1)D()dtd( t ww In view of (5), we have ,1)( n uw and n vw for the Toda lattice. Weights of dependent variables are nonnegative, integer or rational numbers, and independent of . n For example, ),()()( nnn uwuwuw etc. The rank of a monomial is defined as the total weight of the monomial. An ex-pression is uniform in rank if all of its terms have the same rank.
In the first equation of (4), all the monomials have rank 2; in the second equation all the monomials have rank 3. Conversely, requiring uniformity in rank for each equa-tion in (4) allows one to compute the weights of the dependent variables (and thus the scaling symmetry) with elementary linear algebra. Indeed, ),()(1)(),(1)( nnnnn vwuwvwvwuw (6) yields ,2)(,1)( nn vwuw (7) which is consistent with (5). Dilation symmetries, which are Lie-point symmetries, are common to many lat-tice equations. Polynomial DDEs that do not admit a dilation symmetry can be made scaling invariant by extending the set of dependent variables with auxiliary parameters with appropriate scales. A scalar function )( nn u is a conserved density of (1) if there exists a scalar function ),( nn J u called the associated flux, such that [23] nnt J (8) is satisfied on the solutions of (1). In (8), we used the (forward) difference operator, ,)ID( nnnn JJJJ (9) where D denotes the up-shift (forward or right-shift) operator, ,D nn JJ and I is the identity operator. The operator takes the role of a spatial derivative on the shifted variables as many DDEs arise from discretization of a PDE in variables. Most, but not all, den-sities are polynomial in . n u The first three density-flux pairs [11] for (2) are ,),ln( nnnnn uuJu (10) ,, nnnnn uuJu (11) ).(,21 nnnnnnnnn uuuuJuuu (12) The first four density-flux pairs [22] for (4) are ,),ln( )0()0( nnnn uJv (13) ,, nnnn vJu (14) ,,21 nnnnnn vuJvu (15) .),(31 nnnnnnnnnn vvuuJvvuu (16) The densities in (13)-(16) are uniform of ranks through 3, respectively. The corresponding fluxes are also uniform in rank with ranks through 4, respectively. In general, if in (8) R n rank then ,1rank RJ n since .1)D( t w The various piec-es in (8) are uniform in rank. Since (8) holds on solutions of (1), the conservation law ‘inherits’ the dilation symmetry of (1). Consult [22] for our algorithm to compute polynomial conserved densities and fluxes, where we use (4) to illustrate the steps. Non-polynomial densities (which are rare) can be computed by hand or with the method given in [8].
A vector function )( n uG is called a generalized (higher-order) symmetry of (1) if the infinitesimal transformation Guu nn leaves (1) invariant up to order . Consequently, G must satisfy [23] ])[(D t GuFG n (17) on solutions of (1). ])[( GuF n is the Fréchet derivative of F in the direction of . G For the scalar case ),1( N the Fréchet derivative in the direction of G is com-puted as ,D|)(])[( GuFGuFGuF kk knnn (18) which defines the Fréchet derivative operator .D)( kk knn uFuF (19) For the vector case with two components n u and , n v the Fréchet derivative op-erator is .DD DD)(
22 11 k kknk kkn k kknk kknn vFuF vFuF uF (20) Applied to ,),( T21 GG G where T is transpose, one gets .2,1,DD])[( iGvFGuFF kk kn ikk kn ini Gu (21) In (18) - (21), summation is over all positive and negative shifts (including the term without shift, i.e., k For k times).(D...DDD k k Similarly, for k the down-shift operator -1 D is applied repeatedly. The generalization of (20) to N com-ponents should be obvious. The first two symmetries [11] of (2) are ),( nnn uuuG (22) .)()( nnnnnnnnnn uuuuuuuuuuG (23) These symmetries are uniform in rank (rank 2 and 3, respectively). Symmetries of ranks 0 and 1 are both zero. The first two non-trivial symmetries [24] of (4), ,)( nnn nn uuv vvG (24) ,)( )()( nnnnn nnnnnn vvuuv uuvuuvG (25) are uniform in rank. For example, )2(1 G and .4rank )2(2 G The symmetries of lower ranks are trivial. An algorithm to compute polynomial generalized symmetries is described in de-tail in [24].
3. COMPUTATION OF RECURSION OPERATORS 3.1. Definition
A recursion operator connects symmetries , )()( jsj GG (26) where ...,,2,1 j and s is the gap length. The symmetries are linked consecutively if .1 s This happens in most, but not all, cases. For N component systems, is an NN x matrix operator. The defining equation for [6, 23] is ,0)()()(,D t nnn t uFuFFuF (27) where the bracket , denotes the commutator of operators and the composition of operators. The operator )( n uF was defined in (20). F is the Fréchet derivative of in the direction of . F For the scalar case, the operator is often of the form ),(D)I,,D,)ID(()( -11 nn uVuU (28) and in that case .)D()(D k knkknkk uVFUVuUFF (29) For the vector case and the examples under consideration, the elements of the NN x operator matrix are of the form ).(D)I,,D,)ID(()( -11 nijijnijij VU uu Thus, for the two-component case [7] .)D()D( )(D)(D kn ijk ijijknijk ijij ijijkn ijkkijijkn ijkkij vVFUuVFU VvUFVuUFF (30)
The KvM lattice (2) has recursion operator [7] . I1)ID)((DI)(D I1)ID)(D-D)(DI( nnnnnnn nnnn uuuuuuuu uuuu (31)
The Toda lattice (4) has recursion operator [7] .I1)ID()(IDI I1)ID()(IDI
111 111- nnnnnnn nnnn vuuvuvv vvvu (32)
We will now construct the recursion operator (32) for (4). In this case all the terms in (27) are matrix operators. The construction uses the following steps:
Step 1 (Determine the rank of the recursion operator):
The difference in rank of sym-metries is used to compute the rank of the elements of the recursion operator. Using (7), (24) and (25), .43rank,32rank (2)(1) GG (33) Assuming that , (2)(1) GG we use the formula ,rankrankrank )()1( kjkiij GG (34) to compute a rank matrix associated to the operator .12 01rank (35) Step 2 (Determine the form of the recursion operator): where is a sum of terms involving D.and,I,D -1 The coefficients of these terms are admissible power combinations of and,,, nnnn vvuu (which come from the terms on the right hand sides of (4)), so that all the terms have the correct rank. The maximum up-shift and down-shift operator that should be included can be determined by comparing two con-secutive symmetries. Indeed, if the maximum up-shift in the first symmetry is , pn u and the maximum up-shift in the next symmetry is , rpn u then the associated piece that goes into must have .D,...,D,D r2 The same line of reasoning determines the mini-mum down-shift operator to be included. So, in our example ,)()( )()( (36) with ,I)()( nn ucuc ,ID)( cc D,)( I)()( nnnnnn nnnnnn vcvcucuucuc vcvcucuucuc (37) .I)()( nn ucuc As in the continuous case [10], is a linear combination (with constant coeffi-cients jk c ~ of sums of all suitable products of symmetries and covariants (Fréchet deriva-tives of conserved densities) sandwiching .)ID( Hence, ,)ID(~ )(1)( knjj k jk c G (38) where denotes the matrix outer product, defined as .)ID()ID( )ID()ID()ID( )( 2,1)(2)( 1,1)(2 )( 2,1)(1)( 1,1)(1)( 2,)( 1,1)(2 )(1 knjknj knjknjknknjj GG GGGG (39) Only the pair ),( )0()1( n G can be used, otherwise the ranks in (35) would be exceeded. Using (13) and (21), we compute .I10 )0( nn v (40) Therefore, using (38) and renaming ~ c to , c .I1)ID()(0 I1)ID()(0 nnnn nnn vuuvc vvvc (41) Adding (36) and (41), we obtain . (42) Step 3 (Determine the unknown coefficients):
All the terms in (27) need to be com-puted. Referring to [7] for details, the result is: .1and,1 ,0 ccccccc cccccccccc (43) Substituting these constants into (42) finally gives .I1)ID()(IDI I1)ID()(IDI
111 111- nnnnnnn nnnn vuuvuvv vvvu (44) One can readily verify that )2()1( GG with )1( G in (24) and )2( G in (25).
4. THE MATHEMATICA PACKAGE
To use the code, first load the
Mathematica package
DDERecursionOpera-tor.m using the command ];m".onOperatorDDERecursi["Get :=In[2]
Proceeding with the KvM lattice (2) as an example, call the function
DDERe-cursionOperator (which is part of the package) : &}}} t])n, + u[1 t]u[n, + t]u[n, t]n, + (-u[-1 t}]{n, t],[
Here .)ID( The first part of the output (which we assign to R for later use) is indeed the recursion operator given in (31).
First[%];R :=In[4] Now using the first symmetry, generate the next symmetry by calling the func-tion
GenerateSymmetries (which is also part of the package): t]}n,u[2 t]n,u[1 t]u[n,t]n,u[1 t]u[n,t]n,u[1 t]u[n, t]u[n, t]n,u[-1-t]u[n, t]n,u[-1-t]u[n, t]n,u[-1 t]n,{-u[-2=Out[6] 1][[1]]try,firstsymme,mmetries[RGenerateSy: In[6] t])};1,-u[n-t]1,(u[nt]{u[n,tryfirstsymme: In[5]
22 22
Evaluating the next five symmetries starting from the first one, can be done as follows: ] 5] try,firstsymme ,mmetries[RGenerateSy[TableForm: In[7] Due to the length of the output we do not show this result here. The
Mathemati-ca function TableForm will nicely reformat the output in a tabular form. Our package is available at [15].
5. ADDITIONAL EXAMPLES 5.1. Ablowitz-Ladik (AL) Lattice
The AL lattice [19] ),()2( ),()2( nnnnnnnn nnnnnnnn vvvuvvvv uuvuuuuu (45) is an integrable discretization of the nonlinear Schrödinger (NLS) equation. The two recursion operators [7] computed by our package are: , )1(22)1(21 )1(12)1(11)1( (46) with , IIDI)( I,II I,II I,ID nnnnnnnnnnn nnnnnnnn nnnnnnnn nnnnnnn PuPvuvPvuvu PvPvvvvv PuPuuuuu PvPuvuP (47) and , )2(22)2(21 )2(12)2(11)2( (48) with , IID I,II I,II I,I)(ID nnnnnnn nnnnnnnn nnnnnnnn nnnnnnnnnnn PuPvuvP PvPvvvvv PuPuuuuu PvPuvuvuvuP (49) where nnn vuP and . ID It can be shown that .I )1()2()2()1( The RT lattice [20] is given as ,)(),( nnnnnnnnnn vvuuuuuuvv (50) and the recursion operator found by our package coincides with the one in [20]: .I1)ID()( I)(DDID I1)ID()(IDI nnnnnn nnnnnnn nnnnnnn uvvuuu vuuuuuu uuuvvvv (51)
6. CONCLUDING REMARKS
The existence of a recursion operator is a corner stone in establishing the com-plete integrability of nonlinear DDEs because the recursion operators allows one to compute an infinite sequence of generalized symmetries. Therefore, we presented an algorithm to compute recursion operators of nonli-near DDEs with polynomial terms. The algorithm uses the scaling properties, conserva-tion laws, and generalized symmetries of the DDE, but does not require the knowledge of the bi-Hamiltonian operators. The algorithm has been implemented in
Mathematica , a leading computer algebra system. The package
DDERecursionOperators.m uses
In-variantsSymmetries.m to compute the conservation laws and higher-order symmetries of nonlinear DDEs. The algorithm presented in this paper works for many nonlinear DDEs, includ-ing the Kac-van Moerbeke (Volterra), modified Volterra, and Ablowitz-Ladik lattices, as well as standard and relativistic Toda lattices. However, the algorithm does not allow one to compute recursion operators for lattices due to Blaszak-Marciniak and Belov-Chaltikian (see, e.g., [20] for references). An extension of the algorithm that would cov-er these lattices is under investigation.
Acknowledgements-
This material is based upon work supported by the National Science Foundation (U.S.A.) under Grant No. CCF-0830783. J. A. Sanders, J.-P. Wang, M. Hickman and B. Deconinck are gratefully acknowledged for valuable discussions.
7. REFERENCES
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Mathematica package InvariantsSymmetries.m for the symbolic computation of conservation laws and generalized symmetries of non-linear polynomial PDEs and differential-difference equations. Code is available at http://library.wolfram.com/infocenter/MathSource/570. 15. Ü. Göktaş and W. Hereman, 2010.
Mathematica package DDERecursionOperator.m for the symbolic computation of recursion operators for nonlinear polynomial differen-tial-difference equations. The
Mathematica package is available at http://inside.mines.edu/~whereman/software/DDERecursionOperator.
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