Algorithmic determination of infinite-dimensional symmetry groups for integrable systems in 1 + 1 dimensions

Fuchssteiner, Benno; Ivanov, Stefan; Wiwianka, Waldemar

doi:10.1016/s0895-7177(97)00061-7

Cited by 10 publications

(12 citation statements)

References 6 publications

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“…These two implications have been used in the construction of a master symmetry for a given equation. We refer to [13] and its references for concrete examples.…”

Section: Master Symmetries Of Evolution Equationsmentioning

confidence: 99%

Representations of sl(2, ℂ) in category O and master symmetries

Wang

2015

Theor Math Phys

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In this paper, we first give a short account on the indecomposable sl(2, C) modules in the Bernstein-Gelfand-Gelfand (BGG) category O. We show these modules naturally arise for homogeneous integrable nonlinear evolutionary systems. We then develop an approach to construct master symmetries for such integrable systems. This method naturally enables us to compute the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For the new integrable equations such as a Benjamin-Ono type equation, a new integrable Davey-Stewartson type equation and two different versions of (2+1)-dimensional generalised Volterra Chains, we generate their conserved densities using their master symmetries.

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“…These two implications have been used in the construction of a master symmetry for a given equation. We refer to [13] and its references for concrete examples.…”

Section: Master Symmetries Of Evolution Equationsmentioning

confidence: 99%

Representations of sl(2, ℂ) in category O and master symmetries

Wang

2015

Theor Math Phys

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“…This explains the notion integrable for systems (2) that exhibit a complete collection of conserved quantities/symmetries, see Definition 5.2.20 in Abraham and Marsden (1978). It was therefore quite celebrated when researchers discovered the famous Korteweg-de Vries Equation (5) to be integrable, that is, soluble in a rather explicit and practical sense (Zakharov and Faddeev, 1971;Marsden and Weinstein, 1974;Fuchssteiner et al, 1997). As we now know, integrability also applies to a vast number of other important nonlinear partial differential equations such as, e.g., Gardner's, Burger's, nonlinear Schrödinger, and sine Gordon.…”

Section: Manifolds Flows and Integrabilitymentioning

confidence: 99%

“…Theorem 4.3, relying on on a result in combinatorial algebra (Ma and Racine, 1990), formally justifies this identification. In fact, Fuchssteiner (1992a) revealed the basic properties of usual Hamiltonian systems that relate symmetries to conserved quantities and asserted the complete integrability of so many important nonlinear flows (Zakharov and Faddeev, 1971;Gardner, 1971;Fuchssteiner et al, 1997) to be expressible in algebraic terms only, too. Our main result (Section 5) proves each such abstract generator of Heisenberg's dynamics to be Hamiltonian in this generalized algebraic sense.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Reformulation of Heisenberg's Dynamics

Ziegler

Fuchssteiner

2005

Int J Theor Phys

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A structural similarity between Classical Mechanics (CM) and Quantum Mechanics (QM) was revealed by P.A.M. Dirac in terms of Lie Algebras: while in CM the dynamics is determined by the Lie algebra of Poisson brackets on the manifold of scalar fields for classical position/momentum observables Q/P , QM evolves (in Heisenberg's picture) according to the formally similar Lie algebra of commutator brackets of the corresponding operators:where QP − PQ = ih. A further common framework for comparing CM and QM is the category of symplectic manifolds. Other than previous authors, this paper considers phase space of Heisenberg's picture, i.e., the manifold of pairs of operator observables (Q, P) satisfying commutation relation. On a sufficiently high algebraic level of abstraction-which we believe to be of interest on its own-it turns out that this approach leads to a truly nonlinear yet Hamiltonian reformulation of QM evolution.

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“…For more information on the history of completely integrable systems and recursion operators, see [1,11,13,14,17,21,22,23,43,45,50,52,54,56,57,58,59]. Based on studies of formal symmetries and recursion operators, researchers have compiled lists of integrable PDEs [50,51,59,60].…”

Section: Introductionmentioning

confidence: 99%

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

Baldwin

Hereman

2010

International Journal of Computer Mathematics

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A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear PDE to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1 + 1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.

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Algorithmic determination of infinite-dimensional symmetry groups for integrable systems in 1 + 1 dimensions

Cited by 10 publications

References 6 publications

Representations of sl(2, ℂ) in category O and master symmetries

Representations of sl(2, ℂ) in category O and master symmetries

Nonlinear Reformulation of Heisenberg's Dynamics

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

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