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Reducibility of zero curvature representations with application to recursion operators
 PREPRINT SERIES IN GLOBAL ANALYSIS
, 2003
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On the spectral parameter problem
"... We consider the problem whether a nonparametric zerocurvature representation can be embedded into a oneparameter family within the same Lie algebra. After introducing a computable cohomological obstruction, a method using the recursion operator to incorporate the parameter is discussed. 1 ..."
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Cited by 10 (2 self)
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We consider the problem whether a nonparametric zerocurvature representation can be embedded into a oneparameter family within the same Lie algebra. After introducing a computable cohomological obstruction, a method using the recursion operator to incorporate the parameter is discussed. 1
Recursion operator for the IGSG equation
, 2006
"... In this paper we find the inverse and direct recursion operator for the intrinsic generalized sineGordon equation in any number n> 2 of independent variables. Among the flows generated by the direct operator we identify a higherdimensional analogue of the pmKdV equation. Key words. Submanifolds ..."
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In this paper we find the inverse and direct recursion operator for the intrinsic generalized sineGordon equation in any number n> 2 of independent variables. Among the flows generated by the direct operator we identify a higherdimensional analogue of the pmKdV equation. Key words. Submanifolds of constant sectional curvature, intrinsic generalized sineGordon equation, intrinsic generalized wave equation, generalized pmKdV equation. MS Classification (2005). 35Q53, 37K10, 53C42, 58J70. Introduction. General recursion operators as introduced by Olver [24] are pseudodifferential operators ∑s i=−r fiDi x ◦ hi mapping symmetries to symmetries, thus capable of generating infinite series of them. An important generalization by Guthrie [14] eliminates inherent
Locality of symmetries generated by nonhereditary, inhomogeneous, and timedependent recursion operators: a new application for formal symmetries
, 2003
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COVERINGS AND INTEGRABILITY OF THE GAUSS–MAINARDI–CODAZZI EQUATIONS
, 1998
"... Abstract. Using covering theory approach (zerocurvature representations with the gauge group SL2), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained. ..."
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Abstract. Using covering theory approach (zerocurvature representations with the gauge group SL2), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained.
Symmetry, Conserved Charges, and Lax Representations of Nonlinear Field Equations: A Unified Approach
, 2010
"... Abstract: A certain nonNoetherian connection between symmetry and integrability properties of nonlinear field equations in conservationlaw form is studied. It is shown that the symmetry condition alone may lead, in a rather straightforward way, to the construction of a Lax pair, a doubly infinite ..."
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Abstract: A certain nonNoetherian connection between symmetry and integrability properties of nonlinear field equations in conservationlaw form is studied. It is shown that the symmetry condition alone may lead, in a rather straightforward way, to the construction of a Lax pair, a doubly infinite set of (generally nonlocal) conservation laws, and a recursion operator for symmetries. Applications include the chiral field equation and the selfdual YangMills equation.
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, 2004
"... Abstract. In this paper, we investigate the algebraic and geometric properties of the hyperbolic Toda equations uxy = exp(Ku) associated with nondegenerate symmetrizable matrices K. A hierarchy of analogs to the potential modified Korteweg–de Vries equation ut = uxxx + u 3 x is constructed, and its ..."
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Abstract. In this paper, we investigate the algebraic and geometric properties of the hyperbolic Toda equations uxy = exp(Ku) associated with nondegenerate symmetrizable matrices K. A hierarchy of analogs to the potential modified Korteweg–de Vries equation ut = uxxx + u 3 x is constructed, and its relation with the hierarchy for the Korteweg–de Vries equation Tt = Txxx+TTx is established. Group–theoretic structures for the dispersionless (2+1)dimensional Toda equation uxy = exp(−uzz) are obtained. Geometric properties of the multi–component nonlinear Schrödinger equation type systems Ψt = iΨxx +if(Ψ)Ψ (multi–soliton complexes)