On the Integrability of Non-polynomial Scalar Evolution Equations

Sanders, Jan A.; Wang, Jing Ping

doi:10.1006/jdeq.2000.3782

Cited by 27 publications

(54 citation statements)

References 11 publications

(16 reference statements)

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“…The same result can be proved by extending the symbolic approach used in [SW00] for the (1 + 1)-dimensional case.…”

Section: The Commutative Casesupporting

confidence: 59%

On the structure of (2+1)-dimensional commutative and noncommutative integrable equations

Wang

2006

Journal of Mathematical Physics

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We develop the symbolic representation method to derive the hierarchies of (2+1)-dimensional integrable equations from the scalar Lax operators and to study their properties globally. The method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in C or a noncommutative associative algebra. We prove that these hierarchies are indeed quasi-local in the commutative case as conjectured by Mikhailov and Yamilov in 1998, [MY98]. We propose a ring extension based on the symbolic representation. As examples, we give noncommutative versions of KP, mKP and Boussinesq equations.

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“…The same result can be proved by extending the symbolic approach used in [SW00] for the (1 + 1)-dimensional case.…”

Section: The Commutative Casesupporting

confidence: 59%

On the structure of (2+1)-dimensional commutative and noncommutative integrable equations

Wang

2006

Journal of Mathematical Physics

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“…A rigorous proof of the observation "one symmetry implies infinitely many" was established in [22,5] for a wide class of semilinear scalar evolutionary PDEs u t = u nx + f (u, u x , . .…”

Section: Introductionmentioning

confidence: 99%

Some Symmetry Classifications of Hyperbolic Vector Evolution Equations

Anco¹,

Wolf²

2005

JNMP

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Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several O(N )-invariant classes of hyperbolic equations U tx = f (U, U t , U x ) for an N -component vector U (t, x) are considered. In each class we find all scalinghomogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented.

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“…It is very remarkable that in recent years its adequacy has been corroborated a posteriori via deep classification results, see [17,21,27,28].…”

Section: Definition 2 a Partial Differential Equation (Or A System Ofmentioning

confidence: 91%

Remarks on Undecidability, Incompleteness and the Integrability Problem

Reyes

2010

Int J Theor Phys

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In their seminal paper Undecidability and incompleteness in classical mechanics (Int. J. Theor. Phys. 30:1041-1073, N.C.A. da Costa and F.A. Doria introduced a powerful method for studying the appearance of undecidability and incompleteness in mathematics and theoretical physics. In this work their results are applied to integrability theory. Specifically, it is pointed out that it is not possible to expect the existence of an algorithm able to decide whether a given partial differential equation is integrable or not.

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On the Integrability of Non-polynomial Scalar Evolution Equations

Cited by 27 publications

References 11 publications

On the structure of (2+1)-dimensional commutative and noncommutative integrable equations

On the structure of (2+1)-dimensional commutative and noncommutative integrable equations

Some Symmetry Classifications of Hyperbolic Vector Evolution Equations

Remarks on Undecidability, Incompleteness and the Integrability Problem

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