## Integrable Evolution Equations on Associative Algebras (1997)

Citations: | 46 - 7 self |

### BibTeX

@MISC{Olver97integrableevolution,

author = {Peter J. Olver and Vladimir V. Sokolov},

title = {Integrable Evolution Equations on Associative Algebras },

year = {1997}

}

### OpenURL

### Abstract

This paper surveys the classi cation of integrable evolution equations whose eld variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to associative algebra-valued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the bi-Hamiltonian structures for several examples are found.

### Citations

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Citation Context ...l and deep connections between such Hamiltonian operators and classical Riemannian geometry, this result invites an interesting speculation on the proper form of a noncommutative Riemannian geometry, =-=[6]-=-, that will produce such nonlocal Hamiltonian operators. We begin our paper with a review of the basic facts from the theory of associative algebras. In section 3 we present classification results for... |

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Citation Context ...er associative algebras in any depth. The existence of higher order symmetries has been effectively used to classify integrable evolution equations with commutative field variables, [33], [50], [13], =-=[35]. Higher o-=-rder symmetries typically occur in infinite hierarchies, obtained by successively applying a recursion operator to a trivial "seed" symmetry. (Bakirov, [4], proposes an example of a fourth o... |

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Citation Context ... [27]. Thus, we can seek symmetry reductions of our noncommutative integrable equations, leading to associative algebra-valued counterparts of the classical Painlev'e transcendents P-I, : : : , P-VI, =-=[20]-=-. First of all, the first Painlev'e transcendent P-I appears as a symmetry reduction of the matrix KdV equation (1.3). Namely, the classical Galilean symmetry @ @t \Gamma t @ @x + 1 6 @ @u gives us th... |

419 |
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Citation Context ... algebra A. Three important examples are the algebras of n \Theta n matrices, Clifford algebras, [7], [37], and the group algebras appearing in the representation theory of finite-dimensional groups, =-=[8]-=-. Our results, though, do not depend on A being finitedimensional, and so we could also view A as a suitable operator algebra over, say, Hilbert space. For brevity, the noncommutative multiplication u... |

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Citation Context ...], [49], for additional applications of associative, Jordan, and other types of algebras to ordinary differential equations. A particularly powerful approach to integrability was discovered by Magri, =-=[25]-=-, who showed how equations possessing two distinct compatible Hamiltonian structures have a recursion operator, and corresponding infinite hierarchy of commuting biHamiltonian flows and associated con... |

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Citation Context ... We also introduce the notation C u (v) = [u; v]; A u (v) = 2fu; vg; (2:2) so that C u = L u \Gamma R u ; A u = L u +R u : (2:3) Note that the anti-commutator defines a Jordan algebra structure on A, =-=[21]. Finally,-=- for notational convenience, we introduce the "triple anti-commutator" fu; v; wg = 1 2 (uvw +wvu): (2:4) Note that fu; vg = fu; e; vg, where e is the identity element of A. Each of our assoc... |

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Citation Context ...ion operators. Remarkably, one consequence of our studies is that the first order Hamiltonian operators associated with systems of hydrodynamic type, as considered by Dubrovin and Novikov, [9], [10], =-=[11]-=-, do not naturally generalize to local Hamiltonian operators in noncommutative variables: one is required to append certain nonlocal terms in order to satisfy the Jacobi identity. The resulting noncom... |

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Citation Context ...etries of orders60; however, to date, no-one has been able to rigorously prove that Bakirov's system has no additional higher order symmetries.) Ibragimov, Shabat, Sokolov, and Mikhailov, [18], [19], =-=[28]-=-, [41], introduced the concept of a formal symmetry. Their method was then successfully used to classify several basic types of integrable evolutionary systems. We further refer the reader to the exte... |

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Citation Context ...able recursion operators. Remarkably, one consequence of our studies is that the first order Hamiltonian operators associated with systems of hydrodynamic type, as considered by Dubrovin and Novikov, =-=[9]-=-, [10], [11], do not naturally generalize to local Hamiltonian operators in noncommutative variables: one is required to append certain nonlocal terms in order to satisfy the Jacobi identity. The resu... |

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Citation Context ... finding the desired equations. Symmetry reductions of classical soliton equations lead to ordinary differential equations of Painlev'e type, meaning those whose movable singularities are only poles, =-=[1]-=-. Analogous reductions of our noncommutative integrable systems will therefore lead to new associative algebra-valued ordinary differential equations of Painlev'e type. In particular, we exhibit analo... |

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Citation Context ...dentity. The resulting noncommutative operators have some similarities with the non-local (but commutative) Hamiltonian operators of hydrodynamic type introduced by Mokhov and Ferapontov, [12], [31], =-=[32]-=-. Given the beautiful and deep connections between such Hamiltonian operators and classical Riemannian geometry, this result invites an interesting speculation on the proper form of a noncommutative R... |

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Citation Context ... concept of a formal symmetry. Their method was then successfully used to classify several basic types of integrable evolutionary systems. We further refer the reader to the extensive tables in [28], =-=[29]-=-, [30], for a summary of known examples. Due to the complexity of the computation, some of these classification tables relied on the existence of higher order conserved densities, and hence may not be... |

41 | Existence and nonexistence of solitary wave solutions to higher-order model evolution equations
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Citation Context ...his reduces to the standard fifth order symmetry of the ordinary Korteweg-deVries equation, [35]. In the commutative case, there are three different fifth order integrable polynomial equations, [28], =-=[24]-=-; besides the fifth order KdV equation, these are the Sawada--Kotera equation, [39], and the Kaup--Kupershmidt equation, [22]. We have shown that neither of these has an associative counterpart where ... |

38 |
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Citation Context ...the Jacobi identity. The resulting noncommutative operators have some similarities with the non-local (but commutative) Hamiltonian operators of hydrodynamic type introduced by Mokhov and Ferapontov, =-=[12]-=-, [31], [32]. Given the beautiful and deep connections between such Hamiltonian operators and classical Riemannian geometry, this result invites an interesting speculation on the proper form of a nonc... |

34 |
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Citation Context ...rom with the simpler associative algebras in any depth. The existence of higher order symmetries has been effectively used to classify integrable evolution equations with commutative field variables, =-=[33], [50], [1-=-3], [35]. Higher order symmetries typically occur in infinite hierarchies, obtained by successively applying a recursion operator to a trivial "seed" symmetry. (Bakirov, [4], proposes an exa... |

32 |
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Citation Context ...alytical properties and (real) solutions. 3 Not every scalar integrable system has a noncommutative counterpart. There are no purely matrix analogues of the fifth order Sawada-Kotera, [39], and Kaup, =-=[22]-=-, equations when the right hand side of the evolution equation only involves the field variable u and its derivatives, but we suspect there may be versions that also depend on the transpose u and its ... |

31 |
On Poisson brackets of hydrodynamic type
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Citation Context ...recursion operators. Remarkably, one consequence of our studies is that the first order Hamiltonian operators associated with systems of hydrodynamic type, as considered by Dubrovin and Novikov, [9], =-=[10]-=-, [11], do not naturally generalize to local Hamiltonian operators in noncommutative variables: one is required to append certain nonlocal terms in order to satisfy the Jacobi identity. The resulting ... |

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30 |
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Citation Context ...e generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], [46], [48], [47], Fordy, [2], [3], [14], =-=[15]-=-, Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types of nonassociative structures, such as Jord... |

30 |
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Citation Context ...ffects on its analytical properties and (real) solutions. 3 Not every scalar integrable system has a noncommutative counterpart. There are no purely matrix analogues of the fifth order Sawada-Kotera, =-=[39]-=-, and Kaup, [22], equations when the right hand side of the evolution equation only involves the field variable u and its derivatives, but we suspect there may be versions that also depend on the tran... |

29 |
Orthogonal and Symplectic Clifford Algebras
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Citation Context ...ebras. In this paper, the field variables in our systems will take their values in a linear associative algebra A. Three important examples are the algebras of n \Theta n matrices, Clifford algebras, =-=[7]-=-, [37], and the group algebras appearing in the representation theory of finite-dimensional groups, [8]. Our results, though, do not depend on A being finitedimensional, and so we could also view A as... |

26 |
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Citation Context ... direct analogue of the commutative Hamiltonian operator 2uD x + u x , which is the simplest Hamiltonian operator appearing in commutative Hamiltonian systems of "hydrodynamic type", [9], [1=-=0], [11], [36]-=-, which are first order quasilinear systems of partial differential equations. The operator 2uD x + u x defines the first of four known Hamiltonian structures for the inviscid Burgers' equation u t = ... |

26 | Ordinary dierential equations - Ince - 1956 |

20 |
A symmetry approach to exactly solvable evolution equations
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Citation Context ... simpler associative algebras in any depth. The existence of higher order symmetries has been effectively used to classify integrable evolution equations with commutative field variables, [33], [50], =-=[13], [35]. Hi-=-gher order symmetries typically occur in infinite hierarchies, obtained by successively applying a recursion operator to a trivial "seed" symmetry. (Bakirov, [4], proposes an example of a fo... |

19 |
Classification of integrable evolution equations
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(Show Context)
Citation Context ... of orders60; however, to date, no-one has been able to rigorously prove that Bakirov's system has no additional higher order symmetries.) Ibragimov, Shabat, Sokolov, and Mikhailov, [18], [19], [28], =-=[41]-=-, introduced the concept of a formal symmetry. Their method was then successfully used to classify several basic types of integrable evolutionary systems. We further refer the reader to the extensive ... |

18 |
Hamiltonian perturbation theory and water waves
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Citation Context ...bivector \Theta satisfies the quadratic bracket condition [\Theta; \Theta] = 2 v D` (\Theta) = 0: (6:9) Here [ \Delta ; \Delta ] denotes the noncommutative generalization of the Schouten bracket, cf. =-=[34]-=-, between functional multi-vectors. The functional trivector (6.9) can be effectively computed via the noncommutative version of a standard formula based on the formal evolutionary vector field v D` w... |

18 | On symmetries of evolution equations - Sokolov - 1988 |

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Citation Context ... of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], [46], [48], [47], Fordy, [2], [3], [14], [15], Marchenko, =-=[26]-=-, and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types of nonassociative structures, such as Jordan algebras, Jord... |

17 |
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Citation Context ...from certain special types of nonassociative structures, such as Jordan algebras, Jordan triple systems, and so on. This approach leads to many new examples of integrable matrix and vector equations, =-=[42]-=-, [44], which are becoming the focus of significant research activity, [16], [17]. Our starting point is the fact that most of the interesting examples appearing in these papers arise when the field v... |

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Citation Context ...ield variables, [33], [50], [13], [35]. Higher order symmetries typically occur in infinite hierarchies, obtained by successively applying a recursion operator to a trivial "seed" symmetry. =-=(Bakirov, [4]-=-, proposes an example of a fourth order system with a single sixth order symmetry and no higher symmetries of orders60; however, to date, no-one has been able to rigorously prove that Bakirov's system... |

11 |
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Citation Context ...n non-commutative generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], [46], [48], [47], Fordy, =-=[2]-=-, [3], [14], [15], Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types of nonassociative structu... |

11 |
Extension of the module of invertible transformations. Classification of integrable systems
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Citation Context ...pt of a formal symmetry. Their method was then successfully used to classify several basic types of integrable evolutionary systems. We further refer the reader to the extensive tables in [28], [29], =-=[30]-=-, for a summary of known examples. Due to the complexity of the computation, some of these classification tables relied on the existence of higher order conserved densities, and hence may not be compl... |

10 |
Generalised KdV and MKdV equations associated with symmetric spaces
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Citation Context ...-commutative generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], [46], [48], [47], Fordy, [2], =-=[3]-=-, [14], [15], Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types of nonassociative structures, ... |

10 |
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Citation Context ...to the classification of certain non-commutative generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, =-=[45]-=-, [46], [48], [47], Fordy, [2], [3], [14], [15], Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special t... |

9 |
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Citation Context ... higher symmetries of orders60; however, to date, no-one has been able to rigorously prove that Bakirov's system has no additional higher order symmetries.) Ibragimov, Shabat, Sokolov, and Mikhailov, =-=[18]-=-, [19], [28], [41], introduced the concept of a formal symmetry. Their method was then successfully used to classify several basic types of integrable evolutionary systems. We further refer the reader... |

9 |
Hamiltonian systems of hydrodynamic type and constant curvature metrics
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Citation Context ...cobi identity. The resulting noncommutative operators have some similarities with the non-local (but commutative) Hamiltonian operators of hydrodynamic type introduced by Mokhov and Ferapontov, [12], =-=[31]-=-, [32]. Given the beautiful and deep connections between such Hamiltonian operators and classical Riemannian geometry, this result invites an interesting speculation on the proper form of a noncommuta... |

8 |
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Citation Context ...utative generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], [46], [48], [47], Fordy, [2], [3], =-=[14]-=-, [15], Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types of nonassociative structures, such a... |

8 | The Connection Between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painlevé Type
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Citation Context ... \Gamma 3v x uv x + 3vuvuv x : (4:29) 14 5. Matrix Painlev'e Equations. It is well known that the symmetry reductions of integrable systems are ordinary differential equations of Painlev'e type, [1], =-=[27]-=-. Thus, we can seek symmetry reductions of our noncommutative integrable equations, leading to associative algebra-valued counterparts of the classical Painlev'e transcendents P-I, : : : , P-VI, [20].... |

7 |
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Citation Context ...r symmetries of orders60; however, to date, no-one has been able to rigorously prove that Bakirov's system has no additional higher order symmetries.) Ibragimov, Shabat, Sokolov, and Mikhailov, [18], =-=[19]-=-, [28], [41], introduced the concept of a formal symmetry. Their method was then successfully used to classify several basic types of integrable evolutionary systems. We further refer the reader to th... |

7 |
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Citation Context ... The classical matrix chiral model u xy = fu x ; u \Gamma1 ; u y g = 1 2 \Gamma u x u \Gamma1 u y + u y u \Gamma1 u x \Delta ; (5:7) is one of the most important integrable matrix equations. In [42], =-=[43]-=-, [44], this equation was generalized to the case of arbitrary Jordan triple system. The general scaling symmetry reduction of (5.7) leads to the substitution u = x p y q v(x ff y fi ); fffi 6= 0: Wit... |

5 |
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Citation Context ..." from symmetry analysis to the Painlev'e analysis. This could be allow one to classify matrix ordinary differential equations of Painlev'e type. For example, recently Balandin and the second aut=-=hor, [5]-=-, have shown that, although it does not arise as a reduction of either of the matrix mKdV equations, the matrix P-II equation v 00 = 2v 3 + zv + ff e; where ff is a scalar parameter passes the Painlev... |

4 |
Yamilov, Multi-component integrable systems and nonassociative structures
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Citation Context ... Jordan triple systems, and so on. This approach leads to many new examples of integrable matrix and vector equations, [42], [44], which are becoming the focus of significant research activity, [16], =-=[17]-=-. Our starting point is the fact that most of the interesting examples appearing in these papers arise when the field variables take their value in an associative algebra. Particularly important cases... |

4 |
algebras and generalized KdV equations, Theor
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(Show Context)
Citation Context ... classification of certain non-commutative generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], =-=[46]-=-, [48], [47], Fordy, [2], [3], [14], [15], Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types o... |

3 |
Sokolov V V, Deformations of Jordan triple systems and integrable
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(Show Context)
Citation Context ...ertain special types of nonassociative structures, such as Jordan algebras, Jordan triple systems, and so on. This approach leads to many new examples of integrable matrix and vector equations, [42], =-=[44]-=-, which are becoming the focus of significant research activity, [16], [17]. Our starting point is the fact that most of the interesting examples appearing in these papers arise when the field variabl... |

3 |
algebras and integrable systems
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- 1993
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Citation Context ...ification of certain non-commutative generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], [46], =-=[48]-=-, [47], Fordy, [2], [3], [14], [15], Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types of nona... |

2 |
Matrix generalisation of the modified Korteweg-deVries equation
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Citation Context ...The second version is u t = u xxx + 3uu xx \Gamma 3u xx u \Gamma 6uu x u = u xxx + 3 [u; u xx ] \Gamma 6uu x u; (1:6) which is invariant under only u 7! \Gammau . This equation was first described in =-=[23]-=-, where the Lax pair formulation and the inverse scattering problem were studied. Equation (1.6) does not admit a Miura transformation. Finally, the nonlinear Schrodinger equation has a noncommutative... |

2 |
Generalized Schrödinger equations and
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Citation Context ...ion of certain non-commutative generalizations of classical integrable soliton equations. In the literature, integrable multi-component equations have been considered by Svinolupov, [45], [46], [48], =-=[47]-=-, Fordy, [2], [3], [14], [15], Marchenko, [26], and many others. Svinolupov started the systematic classification of such systems, and observed that many arise from certain special types of nonassocia... |