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On proving integrability

Peter H van der Kamp

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We prove the conjecture, formulated in Foursov M V 2000 Inverse Problem 16 259–74, that the system has polynomial symmetries of order 2k and weight 2k + 2n when α = 2(1 − (k/n)) for any non-negative integer k and any positive integer n. Moreover we prove the existence of infinitely many nonpolynomial symmetries for any α. This demonstrates the use of the implicit function theorem of Sanders and Wang together with the symbolic calculus of Gelfand and Dikii to prove the existence of infinitely many symmetries of evolution equations.


PACS

02.30.Ik Integrable systems

02.30.Jr Partial differential equations

MSC

35Q53 KdV-like equations (Korteweg-de Vries, Burgers, sine-Gordon, sinh-Gordon, etc.) (See also 37K10)

35F25 Initial value problems for nonlinear first-order PDE, nonlinear evolution equations

37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)

34C14 Symmetries, invariants

Subjects

Mathematical physics

Dates

Issue 2 (April 2002)

Received 11 October 2001

Published 7 March 2002



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