Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy

Strachan, Ian A. B.; Zuo, Dafeng

doi:10.1063/1.4935936

Cited by 38 publications

(25 citation statements)

References 27 publications

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“…Other approaches to integrability-the structure of Avalued Lax equations, for example, have not been considered here. Part of such a theory have been constructed by the authors in [23] where an A-valued KP hierarchy is constructed via such A-valued Lax equations and operators. In a different direction, there are many other algebra-valued generalizations of KdV equation, from Jordan algebra to Novikov algebra-valued systems [20,21,24,25].…”

Section: Discussionmentioning

confidence: 99%

“…If one restricts such an equation to the space of commuting matrices, one arrives at the equation U t = U x x x + 6UU x which is identical in form to the original KdV equation but with a matrix-valued, as opposed to a scalar-valued, field (see, for example [15,23,26]). The purpose of this paper is to construct A-valued, where A is a Frobenius algebra, generalizations of integrable systems, starting with those associated to an underlying Frobenius manifold and related dispersionless hierarchies, and extending the ideas to topological quantum field theories and dispersive hierarchies.…”

Section: Introductionmentioning

confidence: 99%

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Frobenius manifolds and Frobenius algebra-valued integrable systems

Strachan

Zuo

2017

Lett Math Phys

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The notion of integrability will often extend from systems with scalarvalued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929-952, 1996), Kontsevich and Manin (Inv Math 124: 313-339, 1996). By specializing this construction, using a fixed Frobenius algebra A, one can arrive at such a theory. More generally, one can apply the same idea to construct an A-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an A-valued modified Camassa-Holm equation is constructed.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Frobenius manifolds and Frobenius algebra-valued integrable systems

Strachan

Zuo

2017

Lett Math Phys

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“…By using the knowledge of algebraic reduction, the classical Painlevé IV equation can be extended to a Frobenius Painlevé IV equation as

({, \begin{matrix} left \\ array q_{0}^{″} = \frac{q_{0} (q_{0}^{'}^{2} + ε q_{1}^{'}^{2}) - 2 ε q_{1} q_{0}^{'} q_{1}^{'}}{2 (q_{0}^{2} - ε q_{1}^{2})} + \frac{3}{2} (q_{0}^{3} + 3 ε q_{0} q_{1}^{2}) + 2 t (q_{0}^{2} + ε q_{1}^{2}) + (a_{2} - a_{0} + \frac{t^{2}}{2}) q_{0} - \frac{a_{1}^{2} q_{0}}{2 (q_{0}^{2} - ε q_{1}^{2})}, \\ array q_{1}^{″} = \frac{- q_{1} (q_{0}^{'}^{2} + ε q_{1}^{'}^{2}) + 2 q_{0} q_{0}^{'} q_{1}^{'}}{2 (q_{0}^{2} - ε q_{1}^{2})} + \frac{3}{2} (3 q_{0}^{2} q_{1} + ε q_{1}^{3}) + 4 t q_{0} q_{1} + (a_{2} - a_{0} + \frac{t^{2}}{2}) q_{1} + \frac{a_{1}^{2} q_{1}}{2 (q_{0}^{2} - ε q_{1}^{2})}, \end{matrix})

…”

Section: Frobenius Painlevé IV Equation and Hamilton Systemunclassified

“…First, we recall the Frobenius KP equation

({, \begin{matrix} leftarray \\ array {(4 u_{t} - 12 u u_{x} - 12 ε v v_{x} - u_{x x x})}_{x} - 3 u_{y y} = 0, \\ array {(4 v_{t} - 12 u v_{x} - 12 u_{x} v - v_{x x x})}_{x} - 3 v_{y y} = 0 . \end{matrix})

…”

Section: From Frobenius Modified Kp Hierarchy To Frobenius Painlevé Imentioning

confidence: 99%

“…[16][17][18] Tsuda introduce some connection between the KP hierarchy and Painlevé equations through the reduced Virasoro constraint and similar reductions. 19 In Strachan and Zuo, 20 a new hierarchy called the Frobenius-valued Kakomtsev-Petviashvili hierarchy, which take values in a maximal commutative subalgebra of gl(m, C), was constructed. Then, we consider the Hirota quadratic equation of the commutative version of extended multicomponent Toda hierarchy in Li and He, 21 which should be useful in Frobenius manifold theory.…”

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Affine Weyl group symmetries of Frobenius Painlevé equations

Wang

2019

Math Methods in App Sciences

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In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.

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Liouville integrability and explicit solutions for a Frobenius‐coupled integrable lattice hierarchy

Yang

2022

Math Methods in App Sciences

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We define a Frobenius-coupled integrable lattice hierarchy including its Lax pair.Then the bi-Hamiltonian structures of this hierarchy are constructed, which show that this hierarchy possesses Liouville integrability. We also obtained N-fold Darboux transformation of the Frobenius-coupled integrable lattice hierarchy. Finally, the explicit solutions after onefold Darboux transformation of this hierarchy are investigated when we choose appropriate seed solutions.

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Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy

Cited by 38 publications

References 27 publications

Frobenius manifolds and Frobenius algebra-valued integrable systems

Frobenius manifolds and Frobenius algebra-valued integrable systems

Affine Weyl group symmetries of Frobenius Painlevé equations

Liouville integrability and explicit solutions for a Frobenius‐coupled integrable lattice hierarchy

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